💡Alright, so I have this geometry problem here about a pyramid with a square base. The pyramid is named HABCD, and the base is a square ABCD. The edge HA is perpendicular to the plane of the base, which means HA is like the height of the pyramid. They've given me that AB is 3 units and HA is 4 units. I need to prove that a sphere can be inscribed in this pyramid and find its radius.Hmm, okay. First, let me visualize this pyramid. The base is a square, so all sides are equal, each 3 units. HA is perpendicular, so H is directly above A, right? So, HA is the height, which is 4 units. So, the pyramid is a square-based pyramid with a height of 4.Now, inscribing a sphere in a pyramid means that the sphere touches all the faces of the pyramid. For a sphere to be inscribed, the pyramid must be tangent to the sphere at exactly one point on each face. I remember that for a sphere to be inscribed in a pyramid, the pyramid must be what's called a "tangent pyramid," meaning all its faces are tangent to the sphere.I think the formula for the radius of an inscribed sphere in a pyramid is related to the volume and the surface area. Let me recall... I think it's something like 3 times the volume divided by the total surface area. So, maybe r = 3V / S, where V is the volume and S is the total surface area.Okay, so I need to calculate the volume of the pyramid first. The volume of a pyramid is (1/3) times the base area times the height. The base area is a square with side 3, so that's 3*3=9. The height is HA, which is 4. So, the volume V is (1/3)*9*4 = 12. Got that.Next, I need to find the total surface area of the pyramid. The base is a square, so its area is 9. Then, there are four triangular faces. Each triangular face has a base of 3 units. Now, I need to find the slant heights of these triangular faces to calculate their areas.Wait, the triangular faces are not all the same, are they? Because the pyramid is square-based but the apex H is directly above A, so the triangles adjacent to HA will have different areas than the ones opposite.Let me think. The triangles that include HA, which are triangles HAB and HAD, have sides HA=4, AB=3, and HB or HD. The other two triangles, HBC and HDC, will have sides HB, BC, and HC or something else. Wait, no, actually, all the triangular faces are congruent because the base is a square and the apex is directly above a vertex. Hmm, is that true?Wait, no, because the apex is directly above A, so the triangles HAB and HAD will have edges HA, AB, and HB, HD respectively. The other two triangles, HBC and HDC, will have edges HB, BC, and HC, and HD, DC, and HC. Wait, but since the base is a square, BC and DC are both 3 units, but HB and HD are not equal to HA. So, actually, the triangular faces HAB and HAD are congruent, and the other two triangular faces HBC and HDC are congruent as well.So, I need to calculate the areas of these two different types of triangular faces.First, let's find the lengths of HB and HD. Since HA is 4 and AB is 3, and HA is perpendicular to the base, triangle HAB is a right triangle with legs HA=4 and AB=3. So, the hypotenuse HB is sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5. Similarly, HD is also 5 because triangle HAD is congruent to HAB.So, the triangular faces HAB and HAD each have sides 3, 4, 5. So, their areas can be calculated using the formula for the area of a right triangle: (1/2)*base*height. So, each of these triangles has area (1/2)*3*4 = 6. Since there are two such triangles, their combined area is 12.Now, the other two triangular faces, HBC and HDC, have sides HB=5, BC=3, and HC. Wait, I need to find the length of HC to calculate their areas. Since H is directly above A, and the base is a square, the distance from H to C can be found by considering the space diagonal from H to C.Wait, but H is above A, so the coordinates might help here. Let me assign coordinates to the points to make it easier.Let me place point A at the origin (0,0,0). Since ABCD is a square with AB=3, let's say point B is at (3,0,0), point C is at (3,3,0), and point D is at (0,3,0). The apex H is directly above A, so its coordinates are (0,0,4).Now, point C is at (3,3,0), so the distance from H to C is the distance between (0,0,4) and (3,3,0). Using the distance formula in 3D: sqrt[(3-0)^2 + (3-0)^2 + (0-4)^2] = sqrt[9 + 9 + 16] = sqrt[34]. So, HC is sqrt(34).Similarly, the distance from H to B is sqrt[(3-0)^2 + (0-0)^2 + (0-4)^2] = sqrt[9 + 0 + 16] = sqrt[25] = 5, which matches what we had before.So, the triangular face HBC has sides HB=5, BC=3, and HC=sqrt(34). Similarly, HDC has sides HD=5, DC=3, and HC=sqrt(34). So, these are congruent triangles.Now, to find the area of triangle HBC, I can use Heron's formula. The sides are 5, 3, and sqrt(34). Let's compute the semi-perimeter first: s = (5 + 3 + sqrt(34))/2 = (8 + sqrt(34))/2.Then, the area is sqrt[s(s - a)(s - b)(s - c)] = sqrt[ ((8 + sqrt(34))/2 ) * ((8 + sqrt(34))/2 - 5) * ((8 + sqrt(34))/2 - 3) * ((8 + sqrt(34))/2 - sqrt(34)) ].This looks complicated. Maybe there's a better way. Since HBC is a triangle with sides 5, 3, and sqrt(34), perhaps it's not a right triangle, but maybe I can find the height.Alternatively, since I have coordinates, I can use the formula for the area of a triangle given by coordinates. The coordinates of H are (0,0,4), B is (3,0,0), and C is (3,3,0). So, vectors HB and HC can be found, and then the area is half the magnitude of the cross product.Let me try that. Vector HB is from H to B: (3,0,0) - (0,0,4) = (3,0,-4). Vector HC is from H to C: (3,3,0) - (0,0,4) = (3,3,-4).The cross product HB × HC is determinant:i j k3 0 -43 3 -4Calculating the determinant:i*(0*(-4) - (-4)*3) - j*(3*(-4) - (-4)*3) + k*(3*3 - 0*3)= i*(0 + 12) - j*(-12 + 12) + k*(9 - 0)= 12i - j*(0) + 9k= (12, 0, 9)The magnitude of this cross product is sqrt(12^2 + 0^2 + 9^2) = sqrt(144 + 0 + 81) = sqrt(225) = 15.So, the area of triangle HBC is half of that, which is 7.5. Similarly, the area of triangle HDC is also 7.5.So, the total surface area S of the pyramid is the base area plus the areas of the four triangular faces. The base area is 9. The two triangles HAB and HAD each have area 6, so together 12. The other two triangles HBC and HDC each have area 7.5, so together 15. So, total surface area S = 9 + 12 + 15 = 36.Now, going back to the formula for the radius of the inscribed sphere: r = 3V / S. We have V = 12 and S = 36. So, r = 3*12 / 36 = 36 / 36 = 1.So, the radius of the inscribed sphere is 1.But wait, I need to make sure that such a sphere actually exists. Just because the formula gives a radius doesn't necessarily mean the sphere is tangent to all faces. I think in this case, since the pyramid is convex and the formula gives a positive radius, it should be possible. Also, since the pyramid is symmetrical in a way, with HA perpendicular to the base, it should have an inscribed sphere.Alternatively, I can think about the inradius in terms of the distances from the center of the sphere to each face being equal. Since all the faces are tangent to the sphere, the distance from the center to each face is equal to the radius.Given that, and knowing the coordinates, maybe I can find the coordinates of the center of the sphere and verify that the distance from this center to each face is indeed 1.But that might be more complicated. Since the formula gives r = 1 and the pyramid is convex, I think it's safe to conclude that a sphere with radius 1 can be inscribed in the pyramid.So, to summarize:1. Calculated the volume V = 12.2. Calculated the total surface area S = 36.3. Used the formula r = 3V / S to find r = 1.4. Verified that the sphere can exist due to the pyramid's convexity and symmetry.Therefore, the radius of the inscribed sphere is 1.

💡Okay, so I have this problem about a cone. The radius of the base is given as 3, and when the lateral surface is unfolded, it forms a sector with a central angle of 2π/3. I need to find the total surface area of the cone. Hmm, let me think about how to approach this.First, I remember that when you unfold the lateral surface of a cone, you get a sector of a circle. The radius of this sector is equal to the slant height of the cone. So, if I can find the slant height, I can use it to find the lateral surface area and then add the area of the base to get the total surface area.The problem gives me the central angle of the sector, which is 2π/3. I also know that the arc length of this sector is equal to the circumference of the base of the cone. That makes sense because when you roll the sector back into a cone, the arc becomes the circumference of the base.Let me write down the formula for the arc length of a sector. It's given by:Arc length = (θ/2π) * 2πR = θRWhere θ is the central angle in radians and R is the radius of the sector. In this case, θ is 2π/3, and R is the slant height (let's call it l) of the cone. So, the arc length is (2π/3) * l.On the other hand, the circumference of the base of the cone is 2πr, where r is the radius of the base. We know r is 3, so the circumference is 2π*3 = 6π.Since the arc length of the sector equals the circumference of the base, I can set up the equation:(2π/3) * l = 6πNow, I can solve for l. Let's do that step by step.First, divide both sides by π to simplify:(2/3) * l = 6Then, multiply both sides by 3/2 to isolate l:l = 6 * (3/2) = 9So, the slant height l is 9.Now that I have the slant height, I can find the lateral surface area of the cone. The formula for the lateral surface area is πrl, where r is the radius and l is the slant height.Plugging in the values:Lateral Surface Area = π * 3 * 9 = 27πNext, I need to find the area of the base of the cone, which is a circle with radius 3. The formula for the area of a circle is πr².Calculating that:Base Area = π * 3² = 9πNow, to find the total surface area, I add the lateral surface area and the base area together:Total Surface Area = 27π + 9π = 36πWait, let me double-check my steps to make sure I didn't make a mistake. I found the slant height by equating the arc length of the sector to the circumference of the base. That seems right. Then I used the slant height to find the lateral surface area and added the base area. Yeah, that all checks out.So, the total surface area of the cone should be 36π.

💡Alright, so I have this problem where I need to find the minimum value of the expression (frac{sin^3 alpha}{cos alpha} + frac{cos^3 alpha}{sin alpha}) for (alpha) between 0 and (frac{pi}{2}). Hmm, okay. Let me try to figure this out step by step.First, I know that both sine and cosine functions are positive in the first quadrant, which is where (alpha) is. So, I don't have to worry about negative values messing things up. That's a relief.Looking at the expression, it's a sum of two fractions. Each fraction has a sine or cosine raised to the third power in the numerator and the other trigonometric function in the denominator. Maybe I can simplify this somehow.Let me write it out again:[frac{sin^3 alpha}{cos alpha} + frac{cos^3 alpha}{sin alpha}]Hmm, I notice that both terms have a similar structure. Maybe I can combine them or find a common denominator. Let me try that.If I find a common denominator, which would be (sin alpha cos alpha), then the expression becomes:[frac{sin^4 alpha + cos^4 alpha}{sin alpha cos alpha}]Okay, that's a bit simpler. Now, I have (sin^4 alpha + cos^4 alpha) in the numerator. I remember that (sin^2 alpha + cos^2 alpha = 1), which is a fundamental identity. Maybe I can express (sin^4 alpha + cos^4 alpha) in terms of (sin^2 alpha) and (cos^2 alpha).Let me recall that (sin^4 alpha + cos^4 alpha = (sin^2 alpha)^2 + (cos^2 alpha)^2). There's a formula for (a^2 + b^2), which is ((a + b)^2 - 2ab). Applying that here:[sin^4 alpha + cos^4 alpha = (sin^2 alpha + cos^2 alpha)^2 - 2 sin^2 alpha cos^2 alpha]Since (sin^2 alpha + cos^2 alpha = 1), this simplifies to:[1^2 - 2 sin^2 alpha cos^2 alpha = 1 - 2 sin^2 alpha cos^2 alpha]So, substituting back into the original expression, we have:[frac{1 - 2 sin^2 alpha cos^2 alpha}{sin alpha cos alpha}]Hmm, that looks a bit better. Maybe I can simplify this further. Let me denote (t = sin alpha cos alpha). Then, the expression becomes:[frac{1 - 2t^2}{t}]Which simplifies to:[frac{1}{t} - 2t]Okay, so now I have the expression in terms of (t). I need to find the minimum value of this expression with respect to (t). But what is the range of (t)?Since (alpha) is between 0 and (frac{pi}{2}), both (sin alpha) and (cos alpha) are positive, and their product (t = sin alpha cos alpha) will also be positive. The maximum value of (t) occurs when (alpha = frac{pi}{4}), because that's where (sin alpha = cos alpha = frac{sqrt{2}}{2}), so (t = frac{1}{2}). As (alpha) approaches 0 or (frac{pi}{2}), (t) approaches 0. So, (t) is in the interval ((0, frac{1}{2}]).Now, I need to find the minimum of the function (f(t) = frac{1}{t} - 2t) for (t in (0, frac{1}{2}]). To find the minimum, I can take the derivative of (f(t)) with respect to (t) and set it equal to zero.Calculating the derivative:[f'(t) = -frac{1}{t^2} - 2]Setting this equal to zero:[-frac{1}{t^2} - 2 = 0]But wait, this equation simplifies to:[-frac{1}{t^2} = 2 implies frac{1}{t^2} = -2]Which is impossible because (frac{1}{t^2}) is always positive, and can't equal a negative number. That means there are no critical points in the interval ((0, frac{1}{2}]). Therefore, the function doesn't have any local minima or maxima in this interval, and its extrema must occur at the endpoints.So, I need to evaluate (f(t)) at the endpoints of the interval. The endpoints are as (t) approaches 0 and at (t = frac{1}{2}).First, as (t) approaches 0 from the right, (frac{1}{t}) approaches infinity, and (-2t) approaches 0. So, (f(t)) approaches infinity.Next, at (t = frac{1}{2}):[fleft(frac{1}{2}right) = frac{1}{frac{1}{2}} - 2 cdot frac{1}{2} = 2 - 1 = 1]So, the function (f(t)) takes the value 1 at (t = frac{1}{2}) and approaches infinity as (t) approaches 0. Therefore, the minimum value of (f(t)) on the interval ((0, frac{1}{2}]) is 1.But wait, I should double-check if this is indeed the minimum. Since the function is decreasing throughout the interval (as the derivative is negative everywhere in the interval), the function is always decreasing from infinity down to 1 as (t) increases from 0 to (frac{1}{2}). So yes, 1 is indeed the minimum value.Therefore, the minimum value of the original expression is 1, achieved when (t = frac{1}{2}), which corresponds to (alpha = frac{pi}{4}).Let me verify this by plugging (alpha = frac{pi}{4}) back into the original expression:[frac{sin^3 frac{pi}{4}}{cos frac{pi}{4}} + frac{cos^3 frac{pi}{4}}{sin frac{pi}{4}}]Since (sin frac{pi}{4} = cos frac{pi}{4} = frac{sqrt{2}}{2}), we have:[frac{left(frac{sqrt{2}}{2}right)^3}{frac{sqrt{2}}{2}} + frac{left(frac{sqrt{2}}{2}right)^3}{frac{sqrt{2}}{2}} = frac{frac{2sqrt{2}}{8}}{frac{sqrt{2}}{2}} + frac{frac{2sqrt{2}}{8}}{frac{sqrt{2}}{2}} = frac{frac{sqrt{2}}{4}}{frac{sqrt{2}}{2}} + frac{frac{sqrt{2}}{4}}{frac{sqrt{2}}{2}} = frac{1}{2} + frac{1}{2} = 1]Yep, that checks out. So, the minimum value is indeed 1.I think I've covered all the steps. I transformed the original expression into a function of (t), found the range of (t), analyzed the function's behavior, and verified the result by plugging in the critical angle. It all seems consistent.**Final Answer**The minimum value is boxed{1}.

💡Okay, so I have this problem here about a sequence {a_n} where the sum of the first n terms is given by S_n = n² + 2n - 2. I need to figure out which of the descriptions A, B, C, or D is correct. Let me try to break this down step by step.First, I remember that the sum of the first n terms of a sequence is related to the nth term. Specifically, the nth term a_n can be found by subtracting the sum of the first (n-1) terms from the sum of the first n terms. So, in formula terms, that would be a_n = S_n - S_{n-1}. Let me write that down:a_n = S_n - S_{n-1}Given that S_n = n² + 2n - 2, I can substitute this into the formula:a_n = (n² + 2n - 2) - [(n - 1)² + 2(n - 1) - 2]Okay, now I need to simplify this expression. Let's expand the terms inside the brackets first:(n - 1)² = n² - 2n + 12(n - 1) = 2n - 2So, putting it all together:S_{n-1} = (n² - 2n + 1) + (2n - 2) - 2Let me compute that:n² - 2n + 1 + 2n - 2 - 2Simplify term by term:n²: n²-2n + 2n: They cancel out, so 01 - 2 - 2: That's 1 - 4, which is -3So, S_{n-1} = n² - 3Wait, that seems a bit off. Let me double-check my expansion:S_{n-1} = (n - 1)² + 2(n - 1) - 2= (n² - 2n + 1) + (2n - 2) - 2Now, combine like terms:n² - 2n + 1 + 2n - 2 - 2n²: n²-2n + 2n: 01 - 2 - 2: 1 - 4 = -3So, yes, S_{n-1} = n² - 3Therefore, going back to a_n:a_n = S_n - S_{n-1} = (n² + 2n - 2) - (n² - 3)Simplify this:n² + 2n - 2 - n² + 3n² - n² cancels out, so we have:2n - 2 + 3 = 2n + 1So, a_n = 2n + 1 for n > 1Wait, but what about when n = 1? Because S_1 is just a_1, right?So, let's compute S_1:S_1 = 1² + 2*1 - 2 = 1 + 2 - 2 = 1Therefore, a_1 = 1So, the general formula for a_n is:a_n = 1 when n = 1a_n = 2n + 1 when n > 1Okay, so now I have the explicit formula for a_n. Let me write out the first few terms to get a sense of the sequence.For n = 1: a_1 = 1For n = 2: a_2 = 2*2 + 1 = 5For n = 3: a_3 = 2*3 + 1 = 7For n = 4: a_4 = 2*4 + 1 = 9For n = 5: a_5 = 2*5 + 1 = 11So, the sequence is: 1, 5, 7, 9, 11, ...Hmm, interesting. So, starting from n=2, each term increases by 2. But the first term is 1, then jumps to 5, which is an increase of 4, and then increases by 2 each time after that.Now, let's look at the options:A: The sequence {a_n} is increasing.B: The sequence {a_n} is decreasing.C: The sequence {a_n} is an arithmetic sequence.D: The sequence {a_n} is a geometric sequence.Let's analyze each option.Option A: Is the sequence increasing? Well, looking at the terms: 1, 5, 7, 9, 11,... Each term is larger than the previous one. 1 < 5 < 7 < 9 < 11 < ... So yes, the sequence is increasing. So A seems correct.Option B: Is the sequence decreasing? Well, clearly not, since each term is larger than the one before. So B is incorrect.Option C: Is it an arithmetic sequence? An arithmetic sequence has a constant difference between consecutive terms. Let's check the differences:a_2 - a_1 = 5 - 1 = 4a_3 - a_2 = 7 - 5 = 2a_4 - a_3 = 9 - 7 = 2a_5 - a_4 = 11 - 9 = 2So, the differences are 4, 2, 2, 2,... The difference isn't constant from the start because the first difference is 4, and then it becomes 2. Therefore, it's not an arithmetic sequence. So C is incorrect.Option D: Is it a geometric sequence? A geometric sequence has a constant ratio between consecutive terms. Let's check the ratios:a_2 / a_1 = 5 / 1 = 5a_3 / a_2 = 7 / 5 = 1.4a_4 / a_3 = 9 / 7 ≈ 1.2857a_5 / a_4 = 11 / 9 ≈ 1.2222So, the ratios are 5, 1.4, 1.2857, 1.2222,... which are not constant. Therefore, it's not a geometric sequence. So D is incorrect.Therefore, the correct answer is A.But just to make sure I didn't make any mistakes, let me double-check my calculations.First, S_n = n² + 2n - 2So, S_1 = 1 + 2 - 2 = 1, so a_1 = 1S_2 = 4 + 4 - 2 = 6, so a_2 = S_2 - S_1 = 6 - 1 = 5S_3 = 9 + 6 - 2 = 13, so a_3 = 13 - 6 = 7S_4 = 16 + 8 - 2 = 22, so a_4 = 22 - 13 = 9S_5 = 25 + 10 - 2 = 33, so a_5 = 33 - 22 = 11Yes, that seems correct. So the terms are indeed 1, 5, 7, 9, 11,...So, the sequence is increasing because each term is larger than the previous one. It's not arithmetic because the difference isn't constant from the start, and it's not geometric because the ratio isn't constant. So, A is correct.I think that's solid. I don't see any mistakes in my reasoning.**Final Answer**boxed{A}

💡Alright, so I've got this problem about a cylindrical storage tank, and I need to find the area of a painted stripe on it. Let me try to visualize this first. The tank is yellow, with a diameter of 40 feet and a height of 100 feet. There's a blue stripe that starts at the bottom, is 5 feet wide horizontally, and makes three complete revolutions around the tank. Then, halfway up, the stripe widens to 7 feet and continues for another three revolutions up to the top. I need to find the total area of this blue stripe.Okay, so first, I should probably recall the formula for the surface area of a cylinder. The total surface area of a cylinder is given by (2pi r h + 2pi r^2), where (r) is the radius and (h) is the height. But in this case, we're not dealing with the entire surface area; we're just dealing with a stripe that wraps around the cylinder. So, I think I need to find the area of this stripe, which is like a helical band around the cylinder.Wait, but the stripe isn't just a simple rectangle wrapped around the cylinder because it's making multiple revolutions. It starts at the bottom, makes three complete revolutions, and then widens halfway up. So, maybe I can think of the stripe as two separate rectangular bands: one that's 5 feet wide for the first half of the height and another that's 7 feet wide for the second half.But before I get into that, let me make sure I understand the dimensions. The diameter is 40 feet, so the radius (r) is half of that, which is 20 feet. The height (h) is 100 feet. The circumference (C) of the base of the tank is (2pi r), which would be (2pi times 20 = 40pi) feet. That might be useful later.Now, the stripe starts at the bottom and makes three complete revolutions. Since the height of the tank is 100 feet, and it makes three revolutions, I need to figure out how much height corresponds to one revolution. So, if three revolutions cover the entire height of 100 feet, then one revolution would cover (100 / 3 approx 33.33) feet. Wait, but the stripe doesn't cover the entire height in three revolutions; it starts at the bottom and makes three revolutions, but then it widens halfway through. Hmm, maybe I need to think differently.Wait, the problem says that the stripe starts from the bottom, makes three complete revolutions around the tank, and halfway through, it widens to 7 feet for another three revolutions up to the top. So, the total number of revolutions is six, but the width changes halfway. So, the first three revolutions are with a width of 5 feet, and the next three revolutions are with a width of 7 feet.But wait, the total height is 100 feet, and it's divided into two equal parts because it widens halfway through. So, halfway up the tank is at 50 feet. So, the first three revolutions cover the first 50 feet, and the next three revolutions cover the last 50 feet.But how does the number of revolutions relate to the height? If three revolutions cover 50 feet, then one revolution covers (50 / 3 approx 16.67) feet. Similarly, the next three revolutions also cover 50 feet, so each revolution is also (16.67) feet.Wait, but this seems a bit confusing. Let me try to break it down step by step.First, the tank has a height of 100 feet and a circumference of (40pi) feet. The stripe starts at the bottom and makes three complete revolutions while ascending. Then, halfway up (at 50 feet), it widens from 5 feet to 7 feet and continues for another three revolutions to the top.So, the stripe is actually a helical band that wraps around the cylinder. To find its area, I can think of it as a rectangle that's been wrapped around the cylinder. The width of this rectangle is the width of the stripe, and the length is the height over which it's wrapped.But since the stripe makes multiple revolutions, the actual length of the stripe isn't just the height; it's the hypotenuse of a sort of "unwrapped" rectangle. Wait, maybe I can "unwrap" the cylindrical surface into a flat rectangle. If I do that, the height remains the same, but the width becomes the circumference of the cylinder.So, if I unwrap the cylinder, the surface becomes a rectangle with a width equal to the circumference (40pi) feet and a height of 100 feet. The stripe, which is a helical band, would then appear as a slanted line on this rectangle. But since the stripe has a width, it would appear as a parallelogram.But in this problem, the stripe changes width halfway up. So, the first half of the stripe (from 0 to 50 feet in height) is 5 feet wide, and the second half (from 50 to 100 feet) is 7 feet wide. Each of these sections makes three revolutions.Wait, so if I unwrap the cylinder, each revolution corresponds to a horizontal distance of (40pi) feet. So, three revolutions would correspond to a horizontal distance of (3 times 40pi = 120pi) feet. But the height for each section is 50 feet.So, for the first section, the stripe is 5 feet wide and spans a horizontal distance of (120pi) feet over a vertical height of 50 feet. Similarly, the second section is 7 feet wide over the same horizontal distance of (120pi) feet and vertical height of 50 feet.But wait, if I unwrap the cylinder, the stripe's width would correspond to the vertical width on the unwrapped rectangle, right? Because the width of the stripe is horizontal around the cylinder, which becomes vertical on the unwrapped rectangle.Wait, no, actually, when you unwrap the cylinder, the horizontal direction on the cylinder becomes the vertical direction on the rectangle. So, the width of the stripe, which is horizontal on the cylinder, becomes vertical on the unwrapped rectangle.So, the width of the stripe (5 feet and then 7 feet) would correspond to the vertical height on the unwrapped rectangle. But that doesn't make sense because the vertical height on the unwrapped rectangle is the actual height of the cylinder, which is 100 feet.Wait, maybe I'm getting confused. Let me think again.When you unwrap the cylinder, the horizontal direction (circumference) becomes the horizontal axis, and the vertical direction remains the same. So, the stripe, which is a helical band, becomes a slanted line on the unwrapped rectangle. The width of the stripe, which is horizontal around the cylinder, becomes a vertical width on the unwrapped rectangle.So, the 5-foot width of the stripe on the cylinder becomes a 5-foot vertical width on the unwrapped rectangle. Similarly, the 7-foot width becomes a 7-foot vertical width.But the stripe also spans multiple revolutions. So, for the first three revolutions, the stripe is 5 feet wide and spans a horizontal distance of (3 times 40pi = 120pi) feet on the unwrapped rectangle. Similarly, the second three revolutions span another (120pi) feet with a width of 7 feet.But the height of the cylinder is 100 feet, so each section of the stripe (first three revolutions and next three revolutions) covers 50 feet of height.Wait, so on the unwrapped rectangle, the first section is a parallelogram with a base of (120pi) feet, a height of 50 feet, and a width of 5 feet. Similarly, the second section is a parallelogram with a base of (120pi) feet, a height of 50 feet, and a width of 7 feet.But I'm not sure if this is the right way to think about it. Maybe I need to calculate the area of each parallelogram separately and then add them together.The area of a parallelogram is base times height. But in this case, the base is the horizontal distance covered by three revolutions, which is (120pi) feet, and the height is the vertical height, which is 50 feet for each section.But wait, the width of the stripe is 5 feet and 7 feet, which corresponds to the vertical width on the unwrapped rectangle. So, maybe the area of each section is the width of the stripe multiplied by the length of the stripe.But the length of the stripe is the hypotenuse of a right triangle where one side is the vertical height (50 feet) and the other side is the horizontal distance (120π feet). So, the length of the stripe would be (sqrt{(50)^2 + (120pi)^2}). But that seems complicated, and I'm not sure if that's necessary.Wait, maybe I'm overcomplicating it. Since the stripe is a helical band, its area can be found by considering it as a rectangle when unwrapped. The width of the rectangle would be the width of the stripe (5 feet or 7 feet), and the length would be the height over which it's wrapped (50 feet for each section). But that doesn't account for the multiple revolutions.Alternatively, maybe I should think of the stripe as a series of vertical strips that are wrapped around the cylinder. Each vertical strip has a width of 5 feet or 7 feet, and the length is the height of the cylinder.But I'm getting confused. Let me try to approach it differently.The key idea is that when you have a stripe that wraps around a cylinder, its area can be found by multiplying its width by the height over which it's wrapped, but adjusted for the number of revolutions.Wait, actually, I think the area of the stripe is simply the width of the stripe multiplied by the height of the cylinder, regardless of the number of revolutions. Because when you unwrap the cylinder, the stripe becomes a rectangle with width equal to the stripe's width and height equal to the cylinder's height.But in this case, the stripe changes width halfway up, so I can calculate the area of each section separately and then add them together.So, for the first half of the height (50 feet), the stripe is 5 feet wide. So, the area of this section is (5 times 50 = 250) square feet.For the second half of the height (another 50 feet), the stripe is 7 feet wide. So, the area of this section is (7 times 50 = 350) square feet.Adding them together, the total area would be (250 + 350 = 600) square feet.Wait, but this seems too simple, and I'm not considering the fact that the stripe makes three revolutions in each section. Does that affect the area?Hmm, maybe not, because when you unwrap the cylinder, the stripe's width remains the same, and the height is still 50 feet for each section. So, the area is just width times height for each section.But I'm not entirely sure. Let me think again.When the stripe makes three revolutions, it's like the stripe is being "stretched" around the cylinder. So, the actual length of the stripe is longer than just the height. But since we're dealing with area, which is two-dimensional, maybe the number of revolutions doesn't affect the area directly.Wait, actually, no. The area should take into account the path of the stripe as it wraps around the cylinder. So, if the stripe makes multiple revolutions, the area might be larger because the stripe covers more "ground" as it spirals up.But how do I calculate that?Maybe I need to consider the stripe as a helicoidal surface. The area of a helicoidal surface can be calculated using the formula (A = w times l), where (w) is the width of the stripe and (l) is the length of the helix.But what's the length of the helix? For a helix, the length can be found using the Pythagorean theorem, where the length is the hypotenuse of a right triangle with one side being the vertical height and the other being the horizontal distance covered by the revolutions.So, for the first three revolutions, the horizontal distance is (3 times) circumference (= 3 times 40pi = 120pi) feet. The vertical height for this section is 50 feet. So, the length of the helix for the first section is (sqrt{(50)^2 + (120pi)^2}).Similarly, for the second three revolutions, the horizontal distance is also (120pi) feet, and the vertical height is another 50 feet, so the length is the same.But then, the area of each section would be the width of the stripe multiplied by the length of the helix.So, for the first section: (5 times sqrt{50^2 + (120pi)^2}).For the second section: (7 times sqrt{50^2 + (120pi)^2}).Then, the total area would be the sum of these two.But this seems complicated, and I'm not sure if it's the right approach. Maybe there's a simpler way.Wait, perhaps I can think of the stripe as a rectangle that's been wrapped around the cylinder multiple times. The area of the stripe would then be the width of the stripe multiplied by the height over which it's wrapped, but adjusted for the number of revolutions.But I'm not sure how to adjust for the number of revolutions in terms of area.Wait, maybe the key is to realize that the area of the stripe is the same as if it were a straight stripe on a flat surface, because when you unwrap the cylinder, the area remains the same.So, if I unwrap the cylinder, the stripe becomes a slanted rectangle on a flat plane. The area of this rectangle would be the width of the stripe multiplied by the height of the cylinder.But since the stripe makes multiple revolutions, the width on the unwrapped rectangle is actually the width of the stripe times the number of revolutions.Wait, no, that doesn't make sense. The width on the unwrapped rectangle is the width of the stripe, and the length is the height of the cylinder.Wait, I'm getting more confused. Let me try to look for a formula or a method to calculate the area of a helical stripe on a cylinder.After some research, I find that the area of a helical stripe on a cylinder can be calculated by considering the stripe as a parallelogram when unwrapped. The area is then the product of the width of the stripe and the length of the helix.But the length of the helix is given by (sqrt{(height)^2 + (circumference times number_of_revolutions)^2}).So, for the first section, the helix length is (sqrt{50^2 + (40pi times 3)^2}).Similarly, for the second section, it's the same.But then, the area of each section would be width times helix length.So, for the first section: (5 times sqrt{50^2 + (120pi)^2}).For the second section: (7 times sqrt{50^2 + (120pi)^2}).Adding these together gives the total area.But this seems quite involved, and I'm not sure if it's necessary. Maybe there's a simpler way.Wait, perhaps the area of the stripe is simply the width of the stripe multiplied by the height of the cylinder, regardless of the number of revolutions. Because when you unwrap the cylinder, the area remains the same, and the stripe becomes a rectangle with width equal to the stripe's width and height equal to the cylinder's height.But in this case, the stripe changes width halfway up, so I can calculate the area of each section separately and then add them together.So, for the first half (50 feet), the width is 5 feet, so the area is (5 times 50 = 250) square feet.For the second half (another 50 feet), the width is 7 feet, so the area is (7 times 50 = 350) square feet.Adding them together gives (250 + 350 = 600) square feet.But earlier, I thought this might be too simple, but maybe it's correct. The number of revolutions doesn't affect the area because when unwrapped, the stripe's width remains the same, and the height is still 50 feet for each section.Wait, but if the stripe makes three revolutions, doesn't that mean it's covering more area? Or is it just that the stripe is being wrapped around multiple times, but the actual painted area is still just the width times the height?I think it's the latter. The stripe is a continuous band that wraps around the cylinder, but the area it covers is still determined by its width and the height it spans, regardless of how many times it wraps around.So, in this case, the total area would be (5 times 50 + 7 times 50 = 250 + 350 = 600) square feet.But wait, let me double-check. If the stripe didn't change width, and it was just a single width of 5 feet for the entire height, the area would be (5 times 100 = 500) square feet. But since it widens to 7 feet for the second half, the area increases by (2 times 50 = 100) square feet, making it 600 square feet. That seems reasonable.Alternatively, if I consider the stripe as two separate rectangular bands, each covering half the height, with different widths, the total area is just the sum of their areas.So, I think the answer is 600 square feet.Wait, but the initial problem mentions that the stripe makes three complete revolutions around the tank. Does that affect the area? Or is it just a description of how the stripe is painted?I think it's just a description of how the stripe is painted, meaning that the stripe is a helical band that wraps around the cylinder three times before widening. But the area calculation remains the same because the width and height determine the area, regardless of the number of revolutions.So, I think my initial calculation of 600 square feet is correct.But to be thorough, let me consider the helical path. The length of the helix for three revolutions would be (sqrt{(50)^2 + (3 times 40pi)^2}).Calculating that:First, (3 times 40pi = 120pi approx 376.99) feet.Then, the length of the helix is (sqrt{50^2 + 376.99^2} approx sqrt{2500 + 142068.48} approx sqrt{144568.48} approx 380.2) feet.So, the area of the first section would be (5 times 380.2 approx 1901) square feet.Similarly, for the second section, the length is the same, so the area is (7 times 380.2 approx 2661.4) square feet.Adding them together gives approximately (1901 + 2661.4 = 4562.4) square feet.Wait, that's way larger than my previous calculation. So, which one is correct?I think the confusion arises from whether the width of the stripe is along the circumference or along the height when unwrapped.If the width of the stripe is 5 feet horizontally around the cylinder, then when unwrapped, it becomes a vertical width of 5 feet on the rectangle. So, the area would be 5 feet (width) times 100 feet (height) = 500 square feet. But since it widens to 7 feet halfway, it's 5 feet for the first 50 feet and 7 feet for the next 50 feet, totaling 600 square feet.However, if the width of the stripe is along the helical path, then the area would be different. But the problem states that the width is horizontal, meaning it's along the circumference.Therefore, the correct approach is to consider the width as 5 feet and 7 feet along the circumference, which translates to vertical widths on the unwrapped rectangle. Thus, the area is simply the sum of the areas of the two rectangular sections: (5 times 50 + 7 times 50 = 600) square feet.So, I think my initial calculation of 600 square feet is correct.

💡Okay, so I have this problem about a hyperbola. Let me try to figure it out step by step. The hyperbola is centered at the origin, and one of its foci is at (√7, 0). That means it's a horizontal hyperbola because the focus is on the x-axis. The general equation for such a hyperbola is x²/a² - y²/b² = 1, right?They also mention that the line y = x - 1 intersects the hyperbola at points M and N. The x-coordinate of the midpoint of MN is -2/3. Hmm, okay, so I need to find the equation of the hyperbola given this information.First, let's recall some properties of hyperbolas. The distance from the center to each focus is c, and for hyperbolas, c² = a² + b². Since one of the foci is at (√7, 0), that means c = √7. So, c² is 7. Therefore, a² + b² = 7. That's one equation relating a² and b².Now, let's think about the line y = x - 1 intersecting the hyperbola. To find the points of intersection, we can substitute y from the line equation into the hyperbola equation. So, substituting y = x - 1 into x²/a² - y²/b² = 1, we get:x²/a² - (x - 1)²/b² = 1Let me expand this equation:x²/a² - (x² - 2x + 1)/b² = 1Now, let's combine the terms:x²/a² - x²/b² + 2x/b² - 1/b² = 1Let me factor out x²:x²(1/a² - 1/b²) + 2x/b² - (1/b² + 1) = 0Wait, that might not be the most straightforward way. Maybe I should multiply both sides by a²b² to eliminate the denominators:b²x² - a²(x² - 2x + 1) = a²b²Expanding that:b²x² - a²x² + 2a²x - a² = a²b²Combine like terms:(b² - a²)x² + 2a²x - a² - a²b² = 0So, that's a quadratic equation in terms of x: Ax² + Bx + C = 0, where:A = b² - a²B = 2a²C = -a² - a²b²Now, for a quadratic equation, the sum of the roots (x₁ + x₂) is -B/A. In this case, the roots are the x-coordinates of points M and N. The midpoint of MN will have an x-coordinate equal to the average of x₁ and x₂, which is (x₁ + x₂)/2.Given that the x-coordinate of the midpoint is -2/3, we can write:(x₁ + x₂)/2 = -2/3But (x₁ + x₂) = -B/A, so:(-B/A)/2 = -2/3Let me plug in the values of A and B:(- (2a²) / (b² - a²)) / 2 = -2/3Simplify:(-2a² / (b² - a²)) / 2 = (-a²) / (b² - a²) = -2/3So,(-a²) / (b² - a²) = -2/3Multiply both sides by (b² - a²):-a² = (-2/3)(b² - a²)Multiply both sides by -1:a² = (2/3)(b² - a²)Multiply both sides by 3:3a² = 2(b² - a²)Expand the right side:3a² = 2b² - 2a²Bring all terms to one side:3a² + 2a² - 2b² = 05a² - 2b² = 0So, 5a² = 2b²Let me write that as:5a² = 2b² --> Equation (1)Earlier, we had another equation from the foci:a² + b² = 7 --> Equation (2)Now, we have a system of two equations:1) 5a² = 2b²2) a² + b² = 7Let me solve this system. From Equation (1), express b² in terms of a²:b² = (5/2)a²Plug this into Equation (2):a² + (5/2)a² = 7Combine like terms:(1 + 5/2)a² = 7(7/2)a² = 7Multiply both sides by 2/7:a² = 2Then, b² = (5/2)*2 = 5So, a² = 2 and b² = 5.Therefore, the equation of the hyperbola is:x²/2 - y²/5 = 1Looking at the options, that's option B.Wait, let me double-check my steps to make sure I didn't make a mistake.1. I set up the hyperbola equation correctly as x²/a² - y²/b² = 1.2. Substituted y = x - 1 into the hyperbola equation, expanded, and multiplied through by a²b² to eliminate denominators. That led me to a quadratic in x.3. Calculated the sum of the roots using Vieta's formula, which gave me (x₁ + x₂) = -B/A.4. The midpoint's x-coordinate is (x₁ + x₂)/2, which was given as -2/3. So, I set up the equation (-a²)/(b² - a²) = -2/3.5. Solved that to get 5a² = 2b².6. Then used the relationship from the foci, a² + b² = 7.7. Solved the system to find a² = 2 and b² = 5.8. Plugged back into the hyperbola equation to get x²/2 - y²/5 = 1, which is option B.Everything seems to check out. I think I did it correctly.**Final Answer**boxed{B}