💡Okay, so I have this problem here where I need to figure out the relationship between three expressions: a, b, and c. The variable x is given to be in the interval (e-1, 1). Let me write down what each of these expressions is:- a = ln x- b = 2 ln x- c = ln 3xI need to determine the order of a, b, and c. The options are A, B, C, D, each giving a different order. I need to figure out which one is correct.First, let me recall what the natural logarithm function does. The natural logarithm, ln x, is defined for x > 0. It's a strictly increasing function, which means that as x increases, ln x also increases. However, since x is between e-1 and 1, which is approximately (1.718 - 1, 1) = (0.718, 1), x is a positive number less than 1.Wait, hold on. e is approximately 2.718, so e - 1 is about 1.718. So x is between 1.718 and 1? Wait, that doesn't make sense because 1.718 is greater than 1. So if x is in (e - 1, 1), that would mean x is between approximately 1.718 and 1. But 1.718 is greater than 1, so the interval (1.718, 1) is actually an empty set because 1.718 is greater than 1. That can't be right. Maybe I misread the interval.Wait, maybe it's (e^{-1}, 1). Because e^{-1} is approximately 0.3679, so that would make sense. The interval (0.3679, 1) is valid. Maybe the original problem was written with e^{-1} instead of e - 1. Let me check the original problem again.The original problem says x ∈ (e - 1, 1). Hmm, so it's definitely e - 1, not e^{-1}. So that would mean x is between approximately 1.718 and 1, which is an empty interval because 1.718 > 1. That doesn't make sense. Maybe the interval is written incorrectly? Or perhaps it's a typo.Alternatively, maybe it's (1, e - 1), but that would still be from 1 to approximately 1.718, which is a valid interval. But the original problem says (e - 1, 1), which is from a higher number to a lower number, which is not standard. Usually, intervals are written from lower to higher. So perhaps it's a typo, and it should be (1, e - 1). But without more information, I can't be sure.Wait, maybe I'm overcomplicating. Let me think. If x is in (e - 1, 1), which is approximately (1.718, 1), which is an empty set, that can't be. So maybe the interval is actually (e^{-1}, 1), which is approximately (0.3679, 1). That would make sense because x is between 0.3679 and 1, which is a valid interval where ln x is defined and negative.Alternatively, maybe the problem is correct as written, and I need to interpret it differently. Perhaps it's (e - 1, 1), but since e - 1 is approximately 1.718, and 1 is less than that, so the interval is actually empty. That can't be, because then x wouldn't exist. So perhaps it's a typo, and it's supposed to be (1, e), or (e^{-1}, 1). Given that, I think it's more likely that it's (e^{-1}, 1), because otherwise, the interval is empty.Assuming that, let me proceed with x ∈ (e^{-1}, 1). So x is between approximately 0.3679 and 1. Therefore, ln x is negative because x is less than 1. Since ln 1 = 0, and ln x approaches negative infinity as x approaches 0. So a = ln x is negative.Now, let's compute each expression:- a = ln x- b = 2 ln x = 2a- c = ln 3x = ln 3 + ln x = ln 3 + aSince x is between e^{-1} and 1, ln x is between -1 and 0. So a is between -1 and 0.Let me compute b: since a is negative, 2a is more negative than a. So b < a.Now, c = ln 3 + a. ln 3 is approximately 1.0986. So c is approximately 1.0986 + a. Since a is between -1 and 0, c is between approximately 1.0986 - 1 = 0.0986 and 1.0986 + 0 = 1.0986. So c is positive and between approximately 0.0986 and 1.0986.So, to recap:- a is between -1 and 0- b is between -2 and 0, specifically, since a is between -1 and 0, 2a is between -2 and 0, but since a is greater than -1, 2a is greater than -2. So b is between -2 and 0, but more precisely, since a is between -1 and 0, b is between -2 and 0, but actually, since a is greater than -1, b is greater than -2. So b is between -2 and 0.Wait, but actually, since x is between e^{-1} and 1, ln x is between -1 and 0, so a is between -1 and 0, so 2a is between -2 and 0. So b is between -2 and 0.c is between approximately 0.0986 and 1.0986.So, comparing a, b, and c:- b is between -2 and 0- a is between -1 and 0- c is between approximately 0.0986 and 1.0986So, ordering them from smallest to largest:b < a < cBecause b is the most negative, then a, then c is positive.Therefore, the correct order is b < a < c, which corresponds to option C.Wait, let me double-check. If x is in (e^{-1}, 1), then ln x is between -1 and 0. So a is between -1 and 0, b is 2a, so between -2 and 0, and c is ln 3 + a, which is between ln 3 -1 and ln 3. Since ln 3 is about 1.0986, so c is between approximately 0.0986 and 1.0986.So, b is less than a because b = 2a, and a is negative, so doubling it makes it more negative. Then, c is positive, so it's the largest.Therefore, the order is b < a < c, which is option C.I think that's correct. So the answer should be C.

💡Okay, so I need to figure out the maximum number of planes that can be defined by four parallel lines. Hmm, planes in geometry are defined by three non-collinear points, right? So, if I have four parallel lines, each line is like a straight path that never ends and never meets the others because they're parallel.First, let me visualize this. Imagine four railroad tracks that are all parallel to each other. Each track is a line. Now, if I pick any three of these lines, can they define a plane? Well, since they're parallel, they don't intersect, so just having three parallel lines doesn't give me a plane because they don't form a triangle or anything. So, I need something else to help define the planes.Maybe I can introduce another line that intersects all four of these parallel lines. If I have a line that crosses each of the four parallel lines, that should give me points of intersection. Let's say I have a line that's perpendicular to the four parallel lines. This line would intersect each of them at a single point. So now, I have four points where this perpendicular line meets the parallel lines.With these four points, I can try to form planes. Each plane needs three non-collinear points. So, how many different sets of three points can I make from these four? Well, that's a combination problem. The number of ways to choose 3 points out of 4 is calculated by the combination formula C(n, k) = n! / (k!(n - k)!). Plugging in the numbers, C(4, 3) = 4! / (3!1!) = (4 × 3 × 2 × 1) / ((3 × 2 × 1)(1)) = 24 / 6 = 4. So, there are 4 different planes I can make by choosing any three of these four points.But wait, is that all? Maybe I can get more planes by using different auxiliary lines. If I use another line that intersects the four parallel lines but isn't perpendicular, will that give me different points? Let's say I have another line that's not perpendicular but still intersects all four parallel lines. This would give me four new points, right? So, now I have two sets of four points each, one from each auxiliary line.Now, how many planes can I get from these two sets? For each set of four points, I can make 4 planes, as calculated before. So, that would be 4 planes from the first auxiliary line and 4 planes from the second auxiliary line, totaling 8 planes. But wait, some of these planes might overlap or be the same, especially if the auxiliary lines are parallel or something. Hmm, I need to make sure these auxiliary lines are not parallel to each other to maximize the number of unique planes.If I have two auxiliary lines that are not parallel, they will intersect each other at some point. This intersection point will be common to both sets of four points. So, when I form planes from each set, the planes that include this common point might overlap. That means some of the planes from the first auxiliary line might coincide with planes from the second auxiliary line. So, maybe I'm double-counting some planes here.To avoid that, maybe I should use only one auxiliary line. But then I only get 4 planes. Is there a way to get more than 4 planes without overlapping? Maybe if I use more auxiliary lines that are not parallel to each other or to the original four lines. Let's try adding a third auxiliary line that intersects all four parallel lines and is not parallel to the first two auxiliary lines.Now, with three auxiliary lines, each intersecting the four parallel lines at four distinct points, and none of the auxiliary lines being parallel to each other, I can get more planes. For each auxiliary line, I can form 4 planes, so 3 auxiliary lines would give me 12 planes. But again, some of these planes might overlap because of the common intersection points between the auxiliary lines.This is getting complicated. Maybe there's a better way to think about it. Instead of using auxiliary lines, perhaps I can consider the original four parallel lines and how they can define planes on their own. Since they are parallel, any three of them lie on the same plane, right? Wait, no, because parallel lines don't intersect, so three parallel lines don't define a unique plane. They just lie on infinitely many planes.So, maybe I need to use points from the parallel lines to define planes. If I have four parallel lines, and I pick one point from each line, those four points can define multiple planes depending on how they are arranged. But without any auxiliary lines, it's hard to see how to get multiple planes.Going back to the auxiliary line idea, if I have one auxiliary line intersecting all four parallel lines, I get four points. From these four points, I can form 4 planes by choosing any three points. If I have another auxiliary line, not parallel to the first, intersecting the four parallel lines, I get another four points. From these, I can form another 4 planes. But some of these planes might coincide because the two auxiliary lines intersect at a common point.So, maybe the total number of unique planes is 4 (from the first auxiliary line) plus 3 (from the second auxiliary line, excluding the overlapping plane) plus 2 (from the third auxiliary line, excluding overlapping planes) and so on. This seems similar to the combination formula where the number of planes decreases as we add more auxiliary lines because of overlapping.Wait, maybe it's simpler than that. If I have four parallel lines, and I use one auxiliary line to intersect them, giving me four points, I can form 4 planes. If I use another auxiliary line, not parallel to the first, I get another four points, and from these, I can form 4 more planes, but one of these planes might already exist from the first auxiliary line. So, maybe I get 4 + 3 = 7 planes.But I'm not sure. Maybe I need to think about how many unique planes can be formed by four points in space. Four points can define up to 4 planes if they are in general position, meaning no three are collinear and they don't all lie on the same plane. But in our case, the four points lie on two lines: the four parallel lines and the auxiliary line. So, they are not in general position.Wait, actually, the four points from one auxiliary line are colinear, so they all lie on that auxiliary line. So, they don't form a plane on their own. But when combined with points from another auxiliary line, they can form planes.This is getting a bit tangled. Maybe I should look for a formula or a known result about the number of planes defined by parallel lines and auxiliary lines.I recall that in 3D geometry, the number of planes that can be defined by n lines can vary depending on their configuration. For four parallel lines, the maximum number of planes would be achieved when we have as many different planes as possible, which would involve using auxiliary lines that intersect all four parallel lines and are not parallel to each other.Each auxiliary line can contribute a certain number of planes. If I have k auxiliary lines, each intersecting all four parallel lines, and none of the auxiliary lines are parallel to each other, then each pair of auxiliary lines can define a unique plane. But I'm not sure.Alternatively, maybe the maximum number of planes is determined by the number of ways to choose three lines from the four parallel lines, but since they are parallel, they don't define unique planes. So, perhaps we need to use the auxiliary lines to create points that can define planes.If I have four parallel lines and one auxiliary line intersecting them, I get four points. These four points can define C(4,3) = 4 planes. If I add another auxiliary line, intersecting the four parallel lines at four new points, these can define another 4 planes. However, some of these planes might coincide if the auxiliary lines are arranged in a certain way.To maximize the number of unique planes, the auxiliary lines should be arranged so that the planes they define are all distinct. So, if I have two auxiliary lines, I can get 4 + 4 = 8 planes, but some might overlap. If I have three auxiliary lines, I can get 4 + 4 + 4 = 12 planes, but again, some might overlap.But I think the maximum number of unique planes is actually 4, because each auxiliary line can only define 4 unique planes, and adding more auxiliary lines doesn't necessarily increase the number of unique planes beyond that.Wait, no, that doesn't make sense. If I have two auxiliary lines, each defining 4 planes, but some planes might be the same, so the total number would be less than 8. Similarly, with three auxiliary lines, the total number would be less than 12.But I'm not sure how to calculate the exact number. Maybe I need to think about the configuration where the auxiliary lines are arranged such that the planes they define are all unique.If I have four parallel lines and two auxiliary lines that are not parallel to each other and not parallel to the original lines, then each auxiliary line can define 4 unique planes. Since the auxiliary lines are not parallel, the planes they define won't overlap. So, the total number of planes would be 4 + 4 = 8.But wait, is that correct? If the two auxiliary lines intersect, then the plane defined by their intersection would be the same for both auxiliary lines. So, maybe one plane is overlapping, making the total number 7 instead of 8.Hmm, this is tricky. Maybe I need to consider that each pair of auxiliary lines can define a unique plane, but I'm not sure.Alternatively, perhaps the maximum number of planes is determined by the number of ways to choose three lines from the four parallel lines and the auxiliary lines. But since the four parallel lines are parallel, they don't define unique planes on their own.I think I'm overcomplicating this. Let me try a different approach. If I have four parallel lines, they all lie on infinitely many planes. But to define a unique plane, I need three non-collinear points. So, if I have four parallel lines, and I pick one point from each line, those four points can define multiple planes depending on how they are arranged.If I use one auxiliary line to intersect all four parallel lines, I get four points. From these four points, I can form 4 planes by choosing any three points. If I use another auxiliary line, not parallel to the first, intersecting the four parallel lines, I get another four points. From these, I can form another 4 planes. However, some of these planes might coincide because the two auxiliary lines intersect at a common point.So, the total number of unique planes would be 4 (from the first auxiliary line) plus 3 (from the second auxiliary line, excluding the overlapping plane) plus 2 (from the third auxiliary line, excluding overlapping planes) and so on. This seems like the sum of the first n natural numbers, where n is the number of auxiliary lines.But I'm not sure how many auxiliary lines I can have. In theory, I can have infinitely many auxiliary lines, each intersecting the four parallel lines at different points. But in practice, each auxiliary line can only contribute a certain number of unique planes before overlapping occurs.Wait, maybe the maximum number of planes is actually 4, because each auxiliary line can only define 4 unique planes, and adding more auxiliary lines doesn't necessarily increase the number of unique planes beyond that.But that doesn't seem right. If I have two auxiliary lines, each defining 4 planes, but some overlapping, the total number should be more than 4.I think I need to look for a pattern or a formula. If I have n auxiliary lines, each intersecting the four parallel lines, and none of the auxiliary lines are parallel to each other, then the number of unique planes would be the sum from k=1 to n of (4 - (k - 1)). So, for the first auxiliary line, 4 planes; for the second, 3 new planes; for the third, 2 new planes; and for the fourth, 1 new plane. This gives a total of 4 + 3 + 2 + 1 = 10 planes.But wait, can I have four auxiliary lines? Each auxiliary line intersects the four parallel lines, and none are parallel to each other. So, yes, in theory, I can have four auxiliary lines, each contributing 4, 3, 2, and 1 unique planes respectively, totaling 10 planes.But is 10 the maximum? Or can I get more by using more auxiliary lines? If I have five auxiliary lines, following the same pattern, the fifth auxiliary line would contribute 0 new planes, which doesn't make sense. So, probably, the maximum number is 10.But I'm not entirely sure. Maybe I should verify this with a simpler case. If I have three parallel lines instead of four, how many planes can I get? Using the same logic, with one auxiliary line, I get C(3,3)=1 plane. With two auxiliary lines, I get 1 + 2 = 3 planes. With three auxiliary lines, I get 1 + 2 + 3 = 6 planes. Wait, that doesn't seem right because with three parallel lines, the maximum number of planes should be less.Actually, with three parallel lines, each auxiliary line can define C(3,3)=1 plane. If I have two auxiliary lines, each defining 1 plane, but they might overlap. So, the total number of planes would be 2 if the auxiliary lines are not parallel. But I'm not sure.Maybe my initial approach was wrong. Perhaps the maximum number of planes defined by four parallel lines is actually 4, each defined by one auxiliary line intersecting all four lines. But that seems too low.Wait, no, because each auxiliary line can define multiple planes by choosing different sets of three points. So, with one auxiliary line, 4 planes; with two auxiliary lines, 4 + 4 = 8 planes, but some might overlap; with three auxiliary lines, 4 + 4 + 4 = 12, but more overlaps.But I think the correct approach is to consider that each pair of auxiliary lines can define a unique plane, and the number of such pairs is C(n,2), where n is the number of auxiliary lines. But I'm not sure.I'm getting confused. Maybe I should look up the formula or think about it differently. If I have four parallel lines, they are all coplanar, meaning they lie on the same plane. But I can also have other planes that contain subsets of these lines.Wait, no, four parallel lines don't have to lie on the same plane. They can be arranged in 3D space such that they are all parallel but not coplanar. For example, imagine four railroad tracks that are all parallel but on different levels, like on a multi-level highway. In that case, each pair of lines can lie on a unique plane.But if they are all coplanar, then they all lie on a single plane, and you can't get more planes from them. So, to maximize the number of planes, the four parallel lines should not be coplanar. They should be arranged in 3D space such that no three are coplanar.In that case, each pair of lines can define a unique plane. The number of pairs of four lines is C(4,2)=6. So, does that mean there are 6 planes? But wait, each plane is defined by two lines, but since the lines are parallel, each pair of lines defines the same plane. Wait, no, if the lines are not coplanar, each pair of lines defines a different plane.Wait, no, if two lines are parallel, they define a unique plane. So, if I have four parallel lines, each pair of them defines the same plane if they are coplanar, but if they are not coplanar, each pair defines a different plane.But if they are not coplanar, how can they be parallel? In 3D space, parallel lines can be non-coplanar, meaning they lie on different planes. So, each pair of parallel lines defines a unique plane.Therefore, the number of planes defined by four parallel lines, arranged so that no three are coplanar, would be C(4,2)=6 planes.But wait, that seems conflicting with my earlier thought about using auxiliary lines. Maybe I need to clarify.If the four parallel lines are arranged so that no three are coplanar, then each pair of lines defines a unique plane. So, there are 6 planes. Additionally, if I use auxiliary lines intersecting all four parallel lines, I can get more planes.So, in total, the maximum number of planes would be the 6 planes from the pairs of parallel lines plus the planes defined by the auxiliary lines.But how many planes can I get from the auxiliary lines? If I have one auxiliary line intersecting all four parallel lines, I get four points. From these four points, I can form C(4,3)=4 planes. If I have another auxiliary line, not parallel to the first, intersecting the four parallel lines, I get another four points, and from these, I can form another 4 planes. However, some of these planes might coincide with the ones from the first auxiliary line.But if the auxiliary lines are arranged such that the planes they define are unique, then the total number of planes would be 6 (from the parallel line pairs) plus 4 (from the first auxiliary line) plus 4 (from the second auxiliary line) and so on.But I'm not sure if this is the right way to count. Maybe the planes defined by the auxiliary lines are already included in the 6 planes from the parallel line pairs.Wait, no, because the planes defined by the auxiliary lines are different. They include points from the auxiliary lines, which are not part of the original four parallel lines.So, perhaps the total number of planes is the sum of the planes from the parallel line pairs and the planes from the auxiliary lines.But how many auxiliary lines can I have? In theory, infinitely many, but each can only contribute a finite number of unique planes.This is getting too vague. Maybe I should stick to the initial idea that the maximum number of planes is 4, as each auxiliary line can define 4 planes, and adding more auxiliary lines doesn't necessarily increase the number beyond that.But I'm not confident. I think the correct answer is 4, but I'm not sure. Maybe I should look for a reference or think about it differently.Wait, another approach: each plane is defined by two parallel lines and a point not on them. Since we have four parallel lines, for each pair of lines, we can choose a point from the other two lines to define a plane. So, for each pair of lines, there are C(2,1)=2 choices of points from the remaining two lines. Therefore, the number of planes would be C(4,2)*C(2,1)=6*2=12 planes.But that seems too high. Maybe it's overcounting.Alternatively, for each pair of parallel lines, the plane they define is unique, so that's 6 planes. Additionally, for each line, we can choose a point from another line to define a plane with two other lines. But I'm not sure.I think I'm stuck. Maybe the maximum number of planes is 4, as each auxiliary line can define 4 planes, and adding more auxiliary lines doesn't increase the number beyond that. So, I'll go with 4 as the maximum number of planes.

💡Okay, so I have this problem where a and b are positive real numbers, and they satisfy the equation a + 4b - ab = 0. I need to figure out which of the options A, B, C, or D is correct. Let me try to break this down step by step.First, let me rewrite the given equation to make it easier to work with. The equation is a + 4b - ab = 0. If I move the ab term to the other side, I get a + 4b = ab. Hmm, that looks a bit simpler. Maybe I can express one variable in terms of the other. Let me try solving for a.From a + 4b = ab, I can subtract a from both sides to get 4b = ab - a. Factoring out a on the right side gives me 4b = a(b - 1). Then, solving for a, I get a = (4b)/(b - 1). Okay, so a is expressed in terms of b. Since a and b are positive real numbers, the denominator (b - 1) must be positive, so b must be greater than 1. That's a useful piece of information.Now, let me look at the options one by one.**Option A: The minimum value of ab is 8.**I need to find the minimum value of ab. Since I have a expressed in terms of b, I can substitute that into ab. So, ab = a * b = (4b)/(b - 1) * b = (4b²)/(b - 1). So, ab = 4b²/(b - 1). I need to find the minimum value of this expression for b > 1.To find the minimum, I can take the derivative of ab with respect to b and set it equal to zero. Let me denote f(b) = 4b²/(b - 1). The derivative f’(b) will be [ (8b)(b - 1) - 4b²(1) ] / (b - 1)². Simplifying the numerator: 8b(b - 1) - 4b² = 8b² - 8b - 4b² = 4b² - 8b. So, f’(b) = (4b² - 8b)/(b - 1)².Setting f’(b) = 0, we get 4b² - 8b = 0, which simplifies to 4b(b - 2) = 0. So, b = 0 or b = 2. But since b > 1, we discard b = 0 and take b = 2. Plugging b = 2 back into a = (4b)/(b - 1), we get a = (8)/(1) = 8. So, ab = 8 * 2 = 16. Therefore, the minimum value of ab is 16, not 8. So, Option A is incorrect.**Option B: The minimum value of a + 4b is 16.**Wait, from the original equation, a + 4b = ab. So, if ab has a minimum value of 16, then a + 4b also has a minimum value of 16. That seems straightforward. So, Option B is correct.**Option C: a + 2b ≥ 6 + 4√2.**Hmm, I need to find the minimum value of a + 2b. Let me express a in terms of b again: a = (4b)/(b - 1). So, a + 2b = (4b)/(b - 1) + 2b. Let me combine these terms. To add them, I'll get a common denominator:(4b)/(b - 1) + 2b = [4b + 2b(b - 1)] / (b - 1) = [4b + 2b² - 2b] / (b - 1) = [2b² + 2b] / (b - 1).So, a + 2b = (2b² + 2b)/(b - 1). Let me factor out a 2b from the numerator: 2b(b + 1)/(b - 1). Hmm, not sure if that helps. Maybe I can use calculus again to find the minimum.Let me denote g(b) = (2b² + 2b)/(b - 1). Taking the derivative g’(b):g’(b) = [ (4b + 2)(b - 1) - (2b² + 2b)(1) ] / (b - 1)².Simplifying the numerator:(4b + 2)(b - 1) = 4b² - 4b + 2b - 2 = 4b² - 2b - 2.Subtracting (2b² + 2b):4b² - 2b - 2 - 2b² - 2b = 2b² - 4b - 2.So, g’(b) = (2b² - 4b - 2)/(b - 1)².Setting g’(b) = 0, we get 2b² - 4b - 2 = 0. Dividing both sides by 2: b² - 2b - 1 = 0.Using the quadratic formula: b = [2 ± √(4 + 4)]/2 = [2 ± √8]/2 = [2 ± 2√2]/2 = 1 ± √2.Since b > 1, we take b = 1 + √2. Plugging this back into a = (4b)/(b - 1):a = 4(1 + √2)/(√2) = 4(1 + √2)/√2. Let me rationalize the denominator:4(1 + √2)/√2 = 4(√2 + 2)/2 = 2(√2 + 2) = 2√2 + 4.So, a = 2√2 + 4 and b = 1 + √2. Therefore, a + 2b = (2√2 + 4) + 2(1 + √2) = 2√2 + 4 + 2 + 2√2 = 6 + 4√2. So, the minimum value of a + 2b is indeed 6 + 4√2, which means a + 2b ≥ 6 + 4√2. So, Option C is correct.**Option D: 16/a² + 1/b² ≥ 1/2.**This one looks a bit trickier. Let me see. I need to find the minimum value of 16/a² + 1/b². Since I have a and b related by a + 4b = ab, maybe I can express one variable in terms of the other and then find the minimum.From earlier, a = (4b)/(b - 1). Let me plug this into 16/a² + 1/b²:16/a² + 1/b² = 16/( (4b/(b - 1))² ) + 1/b² = 16/(16b²/(b - 1)²) + 1/b² = (16 * (b - 1)²)/(16b²) + 1/b² = (b - 1)²/b² + 1/b².Simplifying, (b - 1)²/b² + 1/b² = (b² - 2b + 1 + 1)/b² = (b² - 2b + 2)/b² = 1 - 2/b + 2/b².So, 16/a² + 1/b² = 1 - 2/b + 2/b². Let me denote this as h(b) = 1 - 2/b + 2/b².To find the minimum of h(b), I can take the derivative h’(b):h’(b) = 0 + 2/b² - 4/b³.Setting h’(b) = 0:2/b² - 4/b³ = 0.Multiply both sides by b³:2b - 4 = 0 ⇒ 2b = 4 ⇒ b = 2.So, the minimum occurs at b = 2. Plugging back into h(b):h(2) = 1 - 2/2 + 2/(2)² = 1 - 1 + 2/4 = 0 + 0.5 = 0.5.So, the minimum value of 16/a² + 1/b² is 0.5, which is 1/2. Therefore, 16/a² + 1/b² ≥ 1/2. So, Option D is correct.Wait, but let me double-check if this is indeed the minimum. Since h(b) approaches infinity as b approaches 1 from the right, and as b approaches infinity, h(b) approaches 1. So, the minimum at b = 2 is indeed the lowest point. Therefore, Option D is correct.So, summarizing:- Option A: Incorrect (minimum ab is 16)- Option B: Correct (minimum a + 4b is 16)- Option C: Correct (a + 2b ≥ 6 + 4√2)- Option D: Correct (16/a² + 1/b² ≥ 1/2)Therefore, the correct options are B, C, and D.

💡Okay, so I have this equation here: ( z^2 - 16z + 15 = 0 ). The question is asking for the sum of the roots of this equation. Hmm, I remember something about quadratic equations and their roots from my algebra class. Let me try to recall.First, a quadratic equation is generally written as ( az^2 + bz + c = 0 ), where ( a ), ( b ), and ( c ) are coefficients, and ( z ) is the variable. In this case, comparing it to the standard form, I can see that ( a = 1 ), ( b = -16 ), and ( c = 15 ). Okay, that makes sense.Now, I think there's a formula or a property related to the sum and product of the roots of a quadratic equation. I believe it's called Vieta's formulas. Let me try to remember what they say. Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation ( az^2 + bz + c = 0 ), if the roots are ( z_1 ) and ( z_2 ), then:1. The sum of the roots ( z_1 + z_2 = -frac{b}{a} ).2. The product of the roots ( z_1 cdot z_2 = frac{c}{a} ).So, applying this to our equation, the sum of the roots should be ( -frac{b}{a} ). Plugging in the values we have:( z_1 + z_2 = -frac{-16}{1} = 16 ).Wait, that seems straightforward. So the sum of the roots is 16. But just to make sure I'm not making a mistake, let me verify this another way.Another method to find the sum of the roots is to actually solve the quadratic equation and then add the roots together. Let's try that.The quadratic equation is ( z^2 - 16z + 15 = 0 ). To solve for ( z ), I can use the quadratic formula:( z = frac{-b pm sqrt{b^2 - 4ac}}{2a} ).Plugging in the values:( z = frac{-(-16) pm sqrt{(-16)^2 - 4 cdot 1 cdot 15}}{2 cdot 1} ).Simplifying step by step:First, calculate the discriminant ( D = b^2 - 4ac ):( D = (-16)^2 - 4 cdot 1 cdot 15 = 256 - 60 = 196 ).So, the square root of the discriminant is ( sqrt{196} = 14 ).Now, plug this back into the quadratic formula:( z = frac{16 pm 14}{2} ).This gives two solutions:1. ( z = frac{16 + 14}{2} = frac{30}{2} = 15 ).2. ( z = frac{16 - 14}{2} = frac{2}{2} = 1 ).So, the roots are 15 and 1. Adding them together: ( 15 + 1 = 16 ).Okay, that matches what I got using Vieta's formula. So, both methods confirm that the sum of the roots is 16. I think I'm confident with this answer now.Just to recap, Vieta's formula gives a quick way to find the sum of the roots without having to solve the entire equation. It's especially useful when you don't need the actual roots but just their sum or product. In this case, since the equation is quadratic, it's straightforward, but for higher-degree polynomials, Vieta's formulas can be really helpful.I wonder if there's a geometric interpretation of this. If I graph the quadratic equation ( z^2 - 16z + 15 ), it would be a parabola opening upwards because the coefficient of ( z^2 ) is positive. The roots of the equation are the points where the parabola crosses the z-axis. The sum of these roots being 16 tells me something about the symmetry of the parabola.In a quadratic equation, the axis of symmetry is at ( z = -frac{b}{2a} ). Plugging in the values:( z = -frac{-16}{2 cdot 1} = frac{16}{2} = 8 ).So, the axis of symmetry is at ( z = 8 ). This means that the two roots are equally spaced around this line. Since the sum of the roots is 16, and there are two roots, their average is ( frac{16}{2} = 8 ), which aligns with the axis of symmetry. That makes sense because the roots are symmetrically placed around the vertex of the parabola.Let me visualize this. If one root is at 1, the other should be at 15 because 1 and 15 are both 7 units away from 8. So, 8 - 7 = 1 and 8 + 7 = 15. That checks out with the roots we found earlier.This symmetry is a nice property of quadratic equations. It helps in understanding the behavior of the graph without having to plot it. Knowing the axis of symmetry and the sum and product of the roots can give a good idea of how the parabola looks.Another thing I recall is that the product of the roots is ( frac{c}{a} ). In this case, that would be ( frac{15}{1} = 15 ). So, the product of the roots is 15, which also matches our earlier calculation since ( 15 times 1 = 15 ).It's interesting how these formulas tie together different aspects of the quadratic equation. They not only help in finding the roots but also provide insights into the graph's characteristics.I think I've covered the main points here. Using Vieta's formula gives a quick answer, and solving the equation confirms it. Additionally, understanding the symmetry and the relationship between the roots and the coefficients deepens my grasp of quadratic equations.Just to ensure I haven't missed anything, let me summarize:- The equation is ( z^2 - 16z + 15 = 0 ).- Coefficients: ( a = 1 ), ( b = -16 ), ( c = 15 ).- Sum of roots using Vieta's formula: ( -frac{b}{a} = 16 ).- Solving the equation gives roots at 1 and 15, whose sum is 16.- The axis of symmetry is at ( z = 8 ), which is the average of the roots.- The product of the roots is 15, consistent with ( frac{c}{a} ).Everything seems to align correctly. I don't see any mistakes in my reasoning, and both methods lead to the same conclusion. I feel confident that the sum of the roots is indeed 16.

💡Alright, so I have this problem where I need to find the area of a triangle with sides BC=4, AC=6, and AB=8 using Heron's formula. I remember Heron's formula involves something called the semi-perimeter, which is half the sum of all the sides. Let me try to recall the exact formula.First, I think the semi-perimeter, usually denoted as 'p', is calculated by adding all three sides and then dividing by 2. So, if the sides are a, b, and c, then p = (a + b + c)/2. In this case, the sides are 4, 6, and 8. So, p should be (4 + 6 + 8)/2. Let me compute that:4 + 6 is 10, plus 8 is 18. Divided by 2, that's 9. Okay, so p is 9.Now, Heron's formula says that the area of the triangle is the square root of p times (p - a) times (p - b) times (p - c). So, in symbols, it's sqrt[p(p - a)(p - b)(p - c)]. Let me write that down:Area = sqrt[p(p - a)(p - b)(p - c)]Plugging in the values I have:p = 9a = 4b = 6c = 8So, p - a = 9 - 4 = 5p - b = 9 - 6 = 3p - c = 9 - 8 = 1Now, multiply all these together:9 * 5 = 4545 * 3 = 135135 * 1 = 135So, inside the square root, I have 135. Now, I need to find the square root of 135. Hmm, 135 is 9 times 15, right? Because 9*15=135. And the square root of 9 is 3, so sqrt(135) is 3*sqrt(15).Let me double-check my calculations to make sure I didn't make any mistakes. First, the semi-perimeter: (4+6+8)/2=18/2=9. That seems correct. Then, p - a = 5, p - b = 3, p - c =1. Multiplying them together: 9*5=45, 45*3=135, 135*1=135. Square root of 135 is indeed 3*sqrt(15). So, the area is 3*sqrt(15).Looking at the options provided:A: sqrt(15)B: 2*sqrt(15)C: 3*sqrt(15)D: (5/2)*sqrt(15)So, the correct answer should be option C: 3*sqrt(15).Wait, just to make sure, maybe I should consider if Heron's formula applies here. Heron's formula works for any triangle, as long as the sides satisfy the triangle inequality. Let me check if 4, 6, and 8 satisfy the triangle inequality.The sum of any two sides should be greater than the third side.4 + 6 = 10 > 84 + 8 = 12 > 66 + 8 = 14 > 4Yes, all the inequalities hold, so it's a valid triangle. Therefore, Heron's formula is applicable here.Another way to find the area is using the formula (1/2)*base*height, but since I don't know the height, Heron's formula is the way to go here.I think I've covered all the steps and checked my calculations. I'm confident that the area is 3*sqrt(15).

💡Okay, so I need to compute the volume of the solid formed by rotating the astroid given by the parametric equations ( x = a cos^3(t) ) and ( y = a sin^3(t) ) around the line ( y = -a ), where ( a > 0 ). Hmm, let's see. First, I remember that the volume of a solid of revolution can be found using methods like the disk method or the washer method. Since we're rotating around a horizontal line, ( y = -a ), which is below the astroid, I think the washer method might be appropriate here. The washer method formula for the volume when rotating around a horizontal line ( y = c ) is:[V = pi int_{a}^{b} left[ (f(t) - c)^2 - (g(t) - c)^2 right] cdot frac{dx}{dt} dt]But wait, in this case, the astroid is given parametrically, so I need to express everything in terms of the parameter ( t ). The parametric equations are:[x = a cos^3(t)][y = a sin^3(t)]So, the first step is to find the derivative ( frac{dx}{dt} ). Let's compute that:[frac{dx}{dt} = frac{d}{dt} [a cos^3(t)] = 3a cos^2(t) (-sin(t)) = -3a cos^2(t) sin(t)]Okay, so ( frac{dx}{dt} = -3a cos^2(t) sin(t) ). Now, since we're rotating around ( y = -a ), the distance from a point ( y ) on the astroid to the line ( y = -a ) is ( y - (-a) = y + a ). But since the astroid is symmetric, I think we can take advantage of that to simplify the integral. The astroid is symmetric about both the x-axis and y-axis, so maybe we can compute the volume for one quadrant and multiply appropriately.Wait, actually, when rotating around ( y = -a ), the entire astroid is above ( y = -a ) because ( y = a sin^3(t) ) ranges from ( -a ) to ( a ). So, the distance from any point on the astroid to ( y = -a ) is ( y + a ), which is always non-negative.But actually, when ( t ) goes from 0 to ( 2pi ), ( y ) goes from ( a ) to ( -a ) and back to ( a ). So, perhaps I need to be careful about the limits of integration.Wait, maybe it's better to consider the entire parametric curve from ( t = 0 ) to ( t = 2pi ). But since the astroid is symmetric, perhaps integrating from 0 to ( pi/2 ) and multiplying by 4 would suffice? Hmm, not sure. Let's think.Alternatively, maybe using the theorem of Pappus would be more straightforward. The volume of a solid of revolution generated by rotating a plane figure about an external axis is equal to the product of the area of the figure and the distance traveled by its centroid.So, if I can find the area of the astroid and the centroid, then compute the distance traveled by the centroid when rotated around ( y = -a ), I can find the volume.But wait, is the astroid a closed curve? Yes, it is. So, maybe Pappus's theorem is applicable here. Let me recall the formula:[V = A cdot d]Where ( A ) is the area of the figure and ( d ) is the distance traveled by the centroid.So, first, I need to find the area ( A ) of the astroid. I remember that the area of an astroid is ( frac{3}{8} pi a^2 ). Let me verify that.The parametric equations are ( x = a cos^3(t) ), ( y = a sin^3(t) ). The area can be computed using the formula:[A = frac{1}{2} int_{0}^{2pi} (x frac{dy}{dt} - y frac{dx}{dt}) dt]Let's compute ( frac{dy}{dt} ):[frac{dy}{dt} = frac{d}{dt} [a sin^3(t)] = 3a sin^2(t) cos(t)]So, plugging into the area formula:[A = frac{1}{2} int_{0}^{2pi} [a cos^3(t) cdot 3a sin^2(t) cos(t) - a sin^3(t) cdot (-3a cos^2(t) sin(t))] dt]Simplify:[A = frac{1}{2} int_{0}^{2pi} [3a^2 cos^4(t) sin^2(t) + 3a^2 cos^2(t) sin^4(t)] dt]Factor out ( 3a^2 ):[A = frac{3a^2}{2} int_{0}^{2pi} [cos^4(t) sin^2(t) + cos^2(t) sin^4(t)] dt]Notice that ( cos^4(t) sin^2(t) + cos^2(t) sin^4(t) = cos^2(t) sin^2(t) (cos^2(t) + sin^2(t)) = cos^2(t) sin^2(t) cdot 1 = cos^2(t) sin^2(t) ).So,[A = frac{3a^2}{2} int_{0}^{2pi} cos^2(t) sin^2(t) dt]Using the identity ( sin^2(t) cos^2(t) = frac{1}{4} sin^2(2t) ), and ( sin^2(2t) = frac{1 - cos(4t)}{2} ).So,[A = frac{3a^2}{2} cdot frac{1}{4} int_{0}^{2pi} frac{1 - cos(4t)}{2} dt = frac{3a^2}{16} int_{0}^{2pi} (1 - cos(4t)) dt]Compute the integral:[int_{0}^{2pi} 1 dt = 2pi][int_{0}^{2pi} cos(4t) dt = 0]So,[A = frac{3a^2}{16} cdot 2pi = frac{3a^2 pi}{8}]Okay, so the area is ( frac{3}{8} pi a^2 ). That matches what I remembered.Now, to apply Pappus's theorem, I need the centroid of the astroid. The centroid coordinates ( (bar{x}, bar{y}) ) can be found using:[bar{x} = frac{1}{A} cdot frac{1}{2} int_{0}^{2pi} x^2 frac{dy}{dt} dt][bar{y} = frac{1}{A} cdot frac{1}{2} int_{0}^{2pi} y^2 frac{dx}{dt} dt]But wait, since the astroid is symmetric about both the x-axis and y-axis, the centroid should be at the origin, right? Because for every point ( (x, y) ), there's a point ( (-x, y) ), ( (x, -y) ), etc., which would average out to zero. So, ( bar{x} = 0 ) and ( bar{y} = 0 ).But wait, if the centroid is at the origin, then when we rotate the astroid around ( y = -a ), the distance traveled by the centroid is the circumference of the circle with radius equal to the distance from the centroid to the axis of rotation.The distance from the centroid (which is at (0,0)) to the line ( y = -a ) is ( |0 - (-a)| = a ). So, the distance traveled by the centroid is ( 2pi a ).Therefore, by Pappus's theorem, the volume is:[V = A cdot d = frac{3}{8} pi a^2 cdot 2pi a = frac{3}{4} pi^2 a^3]Wait, that seems straightforward. But let me double-check because sometimes Pappus's theorem can be tricky if the figure isn't entirely on one side of the axis or if there's overlapping.In this case, the astroid is entirely above ( y = -a ) because the lowest point of the astroid is at ( y = -a ). So, when rotating around ( y = -a ), the entire figure is on one side of the axis, so Pappus's theorem should apply without issues.Alternatively, if I were to use the washer method, I would set up the integral as follows:The volume element when rotating around ( y = -a ) is a washer with outer radius ( R(t) = y(t) + a ) and inner radius ( r(t) = 0 ) because the astroid is a closed curve and there's no hole. Wait, actually, no, because the astroid is being rotated, so each point on the astroid creates a circle around ( y = -a ). So, actually, the radius is ( y(t) + a ), and since the astroid is symmetric, we can integrate over the entire curve.But since the astroid is parametrized, we can express the volume as:[V = pi int_{0}^{2pi} [R(t)]^2 cdot frac{dx}{dt} dt]But wait, no, that's not quite right. The washer method requires integrating with respect to ( x ) or ( y ). Since we have a parametric equation, it's better to use the parametric form.Wait, actually, the formula for the volume when rotating around a horizontal line using parametric equations is:[V = pi int_{t_1}^{t_2} [f(t) - c]^2 cdot frac{dx}{dt} dt]But in this case, since we're rotating around ( y = -a ), the radius is ( y(t) + a ), so:[V = pi int_{0}^{2pi} (y(t) + a)^2 cdot frac{dx}{dt} dt]But wait, that integral might not be correct because ( frac{dx}{dt} ) is negative in some regions, which could cause issues. Alternatively, maybe I should express it in terms of ( dy ) instead.Wait, perhaps it's better to express ( x ) as a function of ( y ) and integrate with respect to ( y ). But since it's parametric, that might complicate things.Alternatively, I can use the method of cylindrical shells. The formula for the volume when rotating around a horizontal line is:[V = 2pi int_{a}^{b} (y - c) cdot x cdot dy]But again, since it's parametric, I need to express ( x ) and ( dy ) in terms of ( t ).Wait, let's try that. The formula for the volume using the shell method when rotating around ( y = -a ) is:[V = 2pi int_{t_1}^{t_2} (y(t) + a) cdot x(t) cdot frac{dy}{dt} dt]But I'm not sure if that's correct. Maybe I should refer back to the standard formula.Actually, the shell method formula when rotating around a horizontal axis ( y = c ) is:[V = 2pi int_{a}^{b} (y - c) cdot x cdot dy]But since ( x ) is a function of ( t ) and ( dy ) is also a function of ( t ), we can write:[V = 2pi int_{t_1}^{t_2} (y(t) - (-a)) cdot x(t) cdot frac{dy}{dt} dt = 2pi int_{0}^{2pi} (y(t) + a) cdot x(t) cdot frac{dy}{dt} dt]But wait, let's check the limits. The astroid is traced out as ( t ) goes from 0 to ( 2pi ), so the limits would be from 0 to ( 2pi ).But I'm not sure if this is the correct approach because the shell method usually requires integrating with respect to the variable perpendicular to the axis of rotation. Since we're rotating around a horizontal line, the shell method would integrate with respect to ( y ), but since ( x ) is a function of ( t ), it's a bit more involved.Alternatively, maybe it's better to stick with the washer method. The washer method involves integrating with respect to ( x ) or ( y ). Since the astroid is given parametrically, perhaps expressing ( y ) as a function of ( x ) is complicated, but maybe we can express ( x ) in terms of ( y ).From the parametric equations:[x = a cos^3(t) implies cos(t) = left( frac{x}{a} right)^{1/3}][y = a sin^3(t) implies sin(t) = left( frac{y}{a} right)^{1/3}]And since ( cos^2(t) + sin^2(t) = 1 ), we have:[left( frac{x}{a} right)^{2/3} + left( frac{y}{a} right)^{2/3} = 1]So, the Cartesian equation of the astroid is:[left( frac{x}{a} right)^{2/3} + left( frac{y}{a} right)^{2/3} = 1]This is the standard form of an astroid. Now, to use the washer method, we can solve for ( x ) in terms of ( y ):[left( frac{x}{a} right)^{2/3} = 1 - left( frac{y}{a} right)^{2/3}][frac{x}{a} = left( 1 - left( frac{y}{a} right)^{2/3} right)^{3/2}][x = a left( 1 - left( frac{y}{a} right)^{2/3} right)^{3/2}]But this seems complicated to integrate. Maybe it's better to stick with parametric equations.Wait, going back to Pappus's theorem, I think that was the correct approach. The volume is the area times the distance traveled by the centroid. Since the centroid is at the origin, and the distance from the origin to ( y = -a ) is ( a ), the distance traveled is ( 2pi a ). So, the volume is:[V = A cdot 2pi a = frac{3}{8} pi a^2 cdot 2pi a = frac{3}{4} pi^2 a^3]That seems correct. But just to be thorough, let me try to set up the integral using the washer method and see if I get the same result.Using the washer method, the volume when rotating around ( y = -a ) is:[V = pi int_{y_{text{min}}}^{y_{text{max}}} [R(y)]^2 dy]Where ( R(y) ) is the distance from the curve to the axis of rotation, which is ( y + a ). However, since the astroid is symmetric, we can integrate from ( y = -a ) to ( y = a ), but we need to express ( x ) as a function of ( y ), which we did earlier:[x = a left( 1 - left( frac{y}{a} right)^{2/3} right)^{3/2}]But in the washer method, we need to consider the outer radius and the inner radius. However, since the astroid is a closed curve, when rotating around ( y = -a ), each horizontal slice at height ( y ) will have an outer radius of ( y + a ) and an inner radius of 0, because there's no hole. Wait, no, actually, the astroid is a single loop, so when rotating around ( y = -a ), each point on the astroid creates a circle with radius ( y + a ). But since the astroid is symmetric, the volume would be generated by the entire curve.Wait, actually, no. The washer method requires integrating the area of the washers, which are the difference between the outer radius squared and the inner radius squared. But in this case, since we're rotating a closed curve around an external axis, the inner radius is zero because there's no hole. So, the volume would be:[V = pi int_{y_{text{min}}}^{y_{text{max}}} [R(y)]^2 dy]But ( R(y) = y + a ), and ( y ) ranges from ( -a ) to ( a ). However, the astroid is symmetric, so we can compute the volume for ( y ) from ( -a ) to ( a ), but we need to express ( x ) in terms of ( y ) to set up the integral. But since ( x ) is a function of ( y ), and the astroid is symmetric, we can express the volume as twice the volume from ( y = 0 ) to ( y = a ).Wait, no, because when rotating around ( y = -a ), the entire astroid contributes to the volume. So, perhaps it's better to express the volume as:[V = pi int_{-a}^{a} (y + a)^2 cdot 2x(y) dy]Where ( 2x(y) ) accounts for the left and right sides of the astroid. But since the astroid is symmetric about the y-axis, ( x(y) ) is the same for positive and negative ( x ). So, the volume becomes:[V = 2pi int_{-a}^{a} (y + a)^2 cdot x(y) dy]But ( x(y) = a left( 1 - left( frac{y}{a} right)^{2/3} right)^{3/2} ), so plugging that in:[V = 2pi int_{-a}^{a} (y + a)^2 cdot a left( 1 - left( frac{y}{a} right)^{2/3} right)^{3/2} dy]This integral looks quite complicated. Maybe it's better to use substitution or switch back to parametric form.Alternatively, since we have the parametric equations, we can express the integral in terms of ( t ). The formula for the volume when rotating around ( y = -a ) using parametric equations is:[V = pi int_{t_1}^{t_2} [R(t)]^2 cdot frac{dx}{dt} dt]But wait, that's not quite right because ( frac{dx}{dt} ) can be negative, which would cause issues with the integral. Instead, perhaps we should use the formula for the volume of revolution using parametric equations, which is:[V = pi int_{t_1}^{t_2} [f(t)]^2 cdot frac{dx}{dt} dt]But in this case, ( f(t) ) is the radius, which is ( y(t) + a ). However, since ( frac{dx}{dt} ) is negative in some regions, we need to take the absolute value or adjust the limits accordingly.Alternatively, maybe we can express the integral as:[V = pi int_{0}^{2pi} (y(t) + a)^2 cdot left| frac{dx}{dt} right| dt]But this might complicate things further.Wait, perhaps it's better to use the formula for the volume generated by a parametric curve when rotated about a line. The general formula is:[V = pi int_{t_1}^{t_2} [f(t) - c]^2 cdot frac{dx}{dt} dt]But in this case, ( f(t) = y(t) ) and ( c = -a ), so:[V = pi int_{0}^{2pi} (y(t) + a)^2 cdot frac{dx}{dt} dt]But since ( frac{dx}{dt} ) is negative in some regions, the integral might not give the correct result. Alternatively, we can split the integral into regions where ( frac{dx}{dt} ) is positive and negative.Wait, actually, when using the washer method with parametric equations, the formula is:[V = pi int_{t_1}^{t_2} [R(t)]^2 cdot frac{dx}{dt} dt]But if ( frac{dx}{dt} ) is negative, it would subtract from the volume, which isn't correct. So, perhaps we need to take the absolute value of ( frac{dx}{dt} ) or adjust the limits of integration.Alternatively, since the astroid is symmetric, we can compute the volume for one quadrant and multiply by 4. Let's try that.Consider the first quadrant, where ( t ) goes from 0 to ( pi/2 ). In this region, ( x ) decreases from ( a ) to 0, and ( y ) increases from 0 to ( a ). So, ( frac{dx}{dt} ) is negative here.But if we compute the volume for the first quadrant and multiply by 4, we can avoid dealing with the negative ( frac{dx}{dt} ). So, let's set up the integral for the first quadrant:[V_{text{quad}} = pi int_{0}^{pi/2} (y(t) + a)^2 cdot left| frac{dx}{dt} right| dt]Then, the total volume is ( 4V_{text{quad}} ).But let's compute ( V_{text{quad}} ):[V_{text{quad}} = pi int_{0}^{pi/2} (a sin^3 t + a)^2 cdot 3a cos^2 t sin t dt]Simplify:[V_{text{quad}} = 3pi a^3 int_{0}^{pi/2} (sin^3 t + 1)^2 cos^2 t sin t dt]Expanding ( (sin^3 t + 1)^2 ):[(sin^3 t + 1)^2 = sin^6 t + 2sin^3 t + 1]So,[V_{text{quad}} = 3pi a^3 int_{0}^{pi/2} (sin^6 t + 2sin^3 t + 1) cos^2 t sin t dt]This integral seems complicated, but maybe we can use substitution or known integrals.Let me consider substitution. Let ( u = cos t ), then ( du = -sin t dt ). But the integral has ( sin t dt ), so it might not directly help. Alternatively, perhaps express everything in terms of ( sin t ).Let me denote ( s = sin t ), then ( ds = cos t dt ). Wait, but we have ( cos^2 t sin t dt ). Let me see:Express ( cos^2 t = 1 - sin^2 t ), so:[V_{text{quad}} = 3pi a^3 int_{0}^{pi/2} (sin^6 t + 2sin^3 t + 1)(1 - sin^2 t) sin t dt]Let me expand this:[(sin^6 t + 2sin^3 t + 1)(1 - sin^2 t) = sin^6 t (1 - sin^2 t) + 2sin^3 t (1 - sin^2 t) + 1(1 - sin^2 t)][= sin^6 t - sin^8 t + 2sin^3 t - 2sin^5 t + 1 - sin^2 t]So, the integral becomes:[V_{text{quad}} = 3pi a^3 int_{0}^{pi/2} [sin^6 t - sin^8 t + 2sin^3 t - 2sin^5 t + 1 - sin^2 t] sin t dt]Simplify term by term:[int_{0}^{pi/2} sin^7 t dt - int_{0}^{pi/2} sin^9 t dt + 2int_{0}^{pi/2} sin^4 t dt - 2int_{0}^{pi/2} sin^6 t dt + int_{0}^{pi/2} sin t dt - int_{0}^{pi/2} sin^3 t dt]Now, we can use known integrals for powers of sine:Recall that:[int_{0}^{pi/2} sin^n t dt = frac{sqrt{pi} Gammaleft( frac{n+1}{2} right)}{2 Gammaleft( frac{n}{2} + 1 right)}]But maybe it's easier to use reduction formulas or known results.For even powers:[int_{0}^{pi/2} sin^{2k} t dt = frac{pi}{2} cdot frac{(2k - 1)!!}{(2k)!!}]For odd powers:[int_{0}^{pi/2} sin^{2k+1} t dt = frac{(2k)!!}{(2k + 1)!!}]Let's compute each integral:1. ( int_{0}^{pi/2} sin^7 t dt ): Here, ( n = 7 ), which is odd. So,[int_{0}^{pi/2} sin^7 t dt = frac{6!!}{7!!} = frac{6 cdot 4 cdot 2}{7 cdot 5 cdot 3 cdot 1} = frac{48}{105} = frac{16}{35}]2. ( int_{0}^{pi/2} sin^9 t dt ): Similarly,[int_{0}^{pi/2} sin^9 t dt = frac{8!!}{9!!} = frac{8 cdot 6 cdot 4 cdot 2}{9 cdot 7 cdot 5 cdot 3 cdot 1} = frac{384}{945} = frac{128}{315}]3. ( 2int_{0}^{pi/2} sin^4 t dt ): ( n = 4 ), even.[int_{0}^{pi/2} sin^4 t dt = frac{pi}{2} cdot frac{3!!}{4!!} = frac{pi}{2} cdot frac{3 cdot 1}{4 cdot 2} = frac{pi}{2} cdot frac{3}{8} = frac{3pi}{16}]So, multiplied by 2: ( frac{3pi}{8} )4. ( -2int_{0}^{pi/2} sin^6 t dt ): ( n = 6 ), even.[int_{0}^{pi/2} sin^6 t dt = frac{pi}{2} cdot frac{5!!}{6!!} = frac{pi}{2} cdot frac{5 cdot 3 cdot 1}{6 cdot 4 cdot 2} = frac{pi}{2} cdot frac{15}{48} = frac{5pi}{32}]So, multiplied by -2: ( -frac{5pi}{16} )5. ( int_{0}^{pi/2} sin t dt ): ( n = 1 ), odd.[int_{0}^{pi/2} sin t dt = 1]6. ( -int_{0}^{pi/2} sin^3 t dt ): ( n = 3 ), odd.[int_{0}^{pi/2} sin^3 t dt = frac{2!!}{3!!} = frac{2}{3 cdot 1} = frac{2}{3}]So, multiplied by -1: ( -frac{2}{3} )Now, putting all these together:[V_{text{quad}} = 3pi a^3 left[ frac{16}{35} - frac{128}{315} + frac{3pi}{8} - frac{5pi}{16} + 1 - frac{2}{3} right]]Let's compute each term:First, compute the constants:1. ( frac{16}{35} - frac{128}{315} ):Convert to common denominator:[frac{16}{35} = frac{144}{315}][frac{144}{315} - frac{128}{315} = frac{16}{315}]2. ( frac{3pi}{8} - frac{5pi}{16} ):Convert to common denominator:[frac{3pi}{8} = frac{6pi}{16}][frac{6pi}{16} - frac{5pi}{16} = frac{pi}{16}]3. ( 1 - frac{2}{3} = frac{1}{3} )So, combining all:[V_{text{quad}} = 3pi a^3 left( frac{16}{315} + frac{pi}{16} + frac{1}{3} right)]Convert ( frac{1}{3} ) to a denominator of 315:[frac{1}{3} = frac{105}{315}]So,[V_{text{quad}} = 3pi a^3 left( frac{16 + 105}{315} + frac{pi}{16} right) = 3pi a^3 left( frac{121}{315} + frac{pi}{16} right)]This seems messy, and I might have made a mistake somewhere because the result from Pappus's theorem was much simpler. Let me check my steps.Wait, when I set up the integral for ( V_{text{quad}} ), I used the washer method, but perhaps I should have used the shell method instead. Alternatively, maybe I made a mistake in setting up the integral.Wait, actually, when using the washer method with parametric equations, the formula is:[V = pi int_{t_1}^{t_2} [R(t)]^2 cdot frac{dx}{dt} dt]But since ( frac{dx}{dt} ) is negative in the first quadrant, the integral would subtract from the volume, which isn't correct. So, perhaps I should take the absolute value of ( frac{dx}{dt} ) or adjust the limits.Alternatively, maybe I should use the shell method, which might be more appropriate here.The shell method formula when rotating around a horizontal line ( y = c ) is:[V = 2pi int_{a}^{b} (y - c) cdot x cdot dy]In this case, ( c = -a ), so:[V = 2pi int_{-a}^{a} (y + a) cdot x cdot dy]But since the astroid is symmetric about the y-axis, ( x ) is the same for ( y ) and ( -y ), so we can compute the integral from 0 to ( a ) and multiply by 2:[V = 4pi int_{0}^{a} (y + a) cdot x cdot dy]Now, express ( x ) in terms of ( y ):From the parametric equations, ( x = a cos^3 t ) and ( y = a sin^3 t ). So, ( sin t = left( frac{y}{a} right)^{1/3} ), and ( cos t = sqrt{1 - sin^2 t} = sqrt{1 - left( frac{y}{a} right)^{2/3}} ).Thus,[x = a left( 1 - left( frac{y}{a} right)^{2/3} right)^{3/2}]So, the integral becomes:[V = 4pi int_{0}^{a} (y + a) cdot a left( 1 - left( frac{y}{a} right)^{2/3} right)^{3/2} dy]This integral still looks complicated, but maybe we can use substitution. Let me set ( u = left( frac{y}{a} right)^{2/3} ), so ( y = a u^{3/2} ), and ( dy = frac{3}{2} a u^{1/2} du ).When ( y = 0 ), ( u = 0 ); when ( y = a ), ( u = 1 ).Substituting into the integral:[V = 4pi int_{0}^{1} (a u^{3/2} + a) cdot a left( 1 - u right)^{3/2} cdot frac{3}{2} a u^{1/2} du]Simplify:[V = 4pi cdot frac{3}{2} a^3 int_{0}^{1} (u^{3/2} + 1) left( 1 - u right)^{3/2} u^{1/2} du][= 6pi a^3 int_{0}^{1} (u^{3/2} + 1) left( 1 - u right)^{3/2} u^{1/2} du][= 6pi a^3 int_{0}^{1} (u^{2} + u^{1/2}) left( 1 - u right)^{3/2} du]This integral can be split into two parts:[V = 6pi a^3 left( int_{0}^{1} u^{2} (1 - u)^{3/2} du + int_{0}^{1} u^{1/2} (1 - u)^{3/2} du right)]These integrals are of the form of the Beta function:[int_{0}^{1} u^{c - 1} (1 - u)^{d - 1} du = B(c, d) = frac{Gamma(c) Gamma(d)}{Gamma(c + d)}]So, for the first integral, ( c = 3 ), ( d = 5/2 ):[int_{0}^{1} u^{2} (1 - u)^{3/2} du = B(3, 5/2) = frac{Gamma(3) Gamma(5/2)}{Gamma(3 + 5/2)} = frac{2! cdot frac{3sqrt{pi}}{4}}{frac{15sqrt{pi}}{8}} = frac{2 cdot frac{3sqrt{pi}}{4}}{frac{15sqrt{pi}}{8}} = frac{6/4}{15/8} = frac{6}{4} cdot frac{8}{15} = frac{48}{60} = frac{4}{5}]Wait, let me double-check that:Gamma(3) = 2! = 2Gamma(5/2) = (3/2)(1/2)√π = (3/4)√πGamma(3 + 5/2) = Gamma(11/2) = (9/2)(7/2)(5/2)(3/2)(1/2)√π = (945/32)√πSo,B(3, 5/2) = (2 * (3/4)√π) / (945/32 √π) = (6/4) / (945/32) = (3/2) * (32/945) = (96/945) = 32/315Wait, that contradicts my earlier calculation. Let me compute it step by step.Gamma(3) = 2Gamma(5/2) = (3/2)(1/2)√π = (3/4)√πGamma(11/2) = (9/2)(7/2)(5/2)(3/2)(1/2)√π = (945/32)√πSo,B(3, 5/2) = Gamma(3)Gamma(5/2)/Gamma(11/2) = (2)(3/4 √π) / (945/32 √π) = (6/4) / (945/32) = (3/2) * (32/945) = 48/945 = 16/315Similarly, for the second integral, ( c = 3/2 ), ( d = 5/2 ):[int_{0}^{1} u^{1/2} (1 - u)^{3/2} du = B(3/2, 5/2) = frac{Gamma(3/2) Gamma(5/2)}{Gamma(4)} = frac{(sqrt{pi}/2)(3sqrt{pi}/4)}{3!} = frac{(3pi/8)}{6} = frac{pi}{16}]Wait, let's compute it properly:Gamma(3/2) = (1/2)√πGamma(5/2) = (3/2)(1/2)√π = (3/4)√πGamma(4) = 3! = 6So,B(3/2, 5/2) = ( (1/2)√π )( (3/4)√π ) / 6 = (3/8 π) / 6 = (3π/8) / 6 = π/16So, putting it all together:[V = 6pi a^3 left( frac{16}{315} + frac{pi}{16} right ) = 6pi a^3 cdot frac{16}{315} + 6pi a^3 cdot frac{pi}{16}][= frac{96pi a^3}{315} + frac{6pi^2 a^3}{16}]Simplify:[frac{96}{315} = frac{32}{105}][frac{6}{16} = frac{3}{8}]So,[V = frac{32pi a^3}{105} + frac{3pi^2 a^3}{8}]This result doesn't match the one from Pappus's theorem, which was ( frac{3}{4} pi^2 a^3 ). Clearly, I made a mistake somewhere. Let me go back.Wait, when I used Pappus's theorem, I assumed the centroid was at the origin, but maybe that's not correct. Let me re-examine the centroid calculation.The centroid coordinates ( (bar{x}, bar{y}) ) for a parametric curve are given by:[bar{x} = frac{1}{A} cdot frac{1}{2} int_{0}^{2pi} x^2 frac{dy}{dt} dt][bar{y} = frac{1}{A} cdot frac{1}{2} int_{0}^{2pi} y^2 frac{dx}{dt} dt]Given the symmetry, ( bar{x} = 0 ) because the astroid is symmetric about the y-axis. But what about ( bar{y} )?Let's compute ( bar{y} ):[bar{y} = frac{1}{A} cdot frac{1}{2} int_{0}^{2pi} y^2 frac{dx}{dt} dt]We already know ( A = frac{3}{8} pi a^2 ).Compute the integral:[int_{0}^{2pi} y^2 frac{dx}{dt} dt = int_{0}^{2pi} (a sin^3 t)^2 (-3a cos^2 t sin t) dt][= -3a^3 int_{0}^{2pi} sin^6 t cos^2 t sin t dt][= -3a^3 int_{0}^{2pi} sin^7 t cos^2 t dt]This integral is over a full period, and due to the odd powers of sine and cosine, it might be zero. Let me check.Actually, ( sin^7 t cos^2 t ) is an odd function around ( t = pi ), so integrating over ( 0 ) to ( 2pi ) would give zero. Therefore, ( bar{y} = 0 ).So, the centroid is indeed at the origin, and Pappus's theorem gives the correct volume as ( frac{3}{4} pi^2 a^3 ).Therefore, despite the complicated integral setup, the correct answer is ( frac{3}{4} pi^2 a^3 ).I think I made a mistake in setting up the integral for the washer method, possibly due to incorrect limits or misapplying the formula. Given that Pappus's theorem gives a straightforward and elegant result, and the centroid calculation confirms it, I should trust that result.So, the volume of the solid formed by rotating the astroid around ( y = -a ) is ( frac{3}{4} pi^2 a^3 ).