💡Alright, so I've got this problem here that involves some algebraic expressions with powers of 'a' and 'b'. It gives me a sequence of equations:- a + b = 1- a² + b² = 3- a³ + b³ = 4- a⁴ + b⁴ = 7- a⁵ + b⁵ = 11And it's asking me to find what a⁹ + b⁹ equals, with the options being 28, 76, 123, or 199.Hmm, okay. So, first, I need to figure out the pattern here. Let me list out the given values:- n=1: 1- n=2: 3- n=3: 4- n=4: 7- n=5: 11Looking at these numbers, I wonder if there's a recurrence relation here. That is, each term is built from the previous terms. Let's check:Starting from n=3:- 4 (n=3) = 3 (n=2) + 1 (n=1) = 4. Yes, that works.- 7 (n=4) = 4 (n=3) + 3 (n=2) = 7. That also works.- 11 (n=5) = 7 (n=4) + 4 (n=3) = 11. Perfect.So, it seems like each term is the sum of the two preceding terms. That is, S(n) = S(n-1) + S(n-2), where S(n) represents aⁿ + bⁿ.So, if that's the case, I can use this recurrence relation to find the subsequent terms up to n=9.Let me write down what I have so far:- S(1) = 1- S(2) = 3- S(3) = 4- S(4) = 7- S(5) = 11Now, let's compute S(6) to S(9) using the recurrence S(n) = S(n-1) + S(n-2):- S(6) = S(5) + S(4) = 11 + 7 = 18- S(7) = S(6) + S(5) = 18 + 11 = 29- S(8) = S(7) + S(6) = 29 + 18 = 47- S(9) = S(8) + S(7) = 47 + 29 = 76So, according to this pattern, a⁹ + b⁹ should be 76.But wait, let me double-check to make sure I didn't make any calculation errors.Starting from S(1) to S(5):1, 3, 4, 7, 11Compute S(6):11 + 7 = 18 ✔️Compute S(7):18 + 11 = 29 ✔️Compute S(8):29 + 18 = 47 ✔️Compute S(9):47 + 29 = 76 ✔️Okay, that seems consistent. So, 76 is the result for a⁹ + b⁹.But just to be thorough, maybe I should verify this another way. Perhaps by finding the actual values of 'a' and 'b' and then computing a⁹ + b⁹ directly.Given that a + b = 1 and a² + b² = 3.I know that (a + b)² = a² + 2ab + b². So, plugging in the known values:1² = 3 + 2ab1 = 3 + 2abSo, 2ab = 1 - 3 = -2Therefore, ab = -1So, now I have a + b = 1 and ab = -1.These are the sums and products of the roots of a quadratic equation. So, the quadratic equation would be x² - (a + b)x + ab = 0, which is x² - x - 1 = 0.So, the roots are:x = [1 ± sqrt(1 + 4)] / 2 = [1 ± sqrt(5)] / 2Therefore, a and b are (1 + sqrt(5))/2 and (1 - sqrt(5))/2.These are actually the golden ratio and its conjugate. Interesting.Now, if I can express aⁿ + bⁿ in terms of the Fibonacci sequence or something similar, since the recurrence relation we found earlier is similar to the Fibonacci sequence.But regardless, since we've already established the recurrence relation, and the calculations up to S(9) = 76 seem consistent, I think 76 is the correct answer.Alternatively, I can compute a⁹ + b⁹ using the values of 'a' and 'b'.Given that a = (1 + sqrt(5))/2 and b = (1 - sqrt(5))/2.Let me compute a⁹ and b⁹ separately and then add them.But computing a⁹ and b⁹ manually would be tedious, but perhaps I can find a pattern or use the recurrence relation.Wait, since we already have S(n) = S(n-1) + S(n-2), and we've computed up to S(9) = 76, I think that's sufficient.Alternatively, I can use the fact that a and b satisfy the equation x² = x + 1, so we can express higher powers in terms of lower powers.For example:a² = a + 1a³ = a² + a = (a + 1) + a = 2a + 1a⁴ = a³ + a² = (2a + 1) + (a + 1) = 3a + 2a⁵ = a⁴ + a³ = (3a + 2) + (2a + 1) = 5a + 3a⁶ = a⁵ + a⁴ = (5a + 3) + (3a + 2) = 8a + 5a⁷ = a⁶ + a⁵ = (8a + 5) + (5a + 3) = 13a + 8a⁸ = a⁷ + a⁶ = (13a + 8) + (8a + 5) = 21a + 13a⁹ = a⁸ + a⁷ = (21a + 13) + (13a + 8) = 34a + 21Similarly, for b:b² = b + 1b³ = b² + b = (b + 1) + b = 2b + 1b⁴ = b³ + b² = (2b + 1) + (b + 1) = 3b + 2b⁵ = b⁴ + b³ = (3b + 2) + (2b + 1) = 5b + 3b⁶ = b⁵ + b⁴ = (5b + 3) + (3b + 2) = 8b + 5b⁷ = b⁶ + b⁵ = (8b + 5) + (5b + 3) = 13b + 8b⁸ = b⁷ + b⁶ = (13b + 8) + (8b + 5) = 21b + 13b⁹ = b⁸ + b⁷ = (21b + 13) + (13b + 8) = 34b + 21Therefore, a⁹ + b⁹ = (34a + 21) + (34b + 21) = 34(a + b) + 42We know that a + b = 1, so:a⁹ + b⁹ = 34(1) + 42 = 34 + 42 = 76So, that confirms it. a⁹ + b⁹ is indeed 76.Therefore, the correct answer is 76, which corresponds to option B.**Final Answer**boxed{B}

💡Okay, so I have this function f(x) = ax + ln x, and there are three parts to the problem. Let me take them one by one.**1. When a = -1, find the maximum value of f(x).**Alright, so first, I need to substitute a = -1 into the function. That gives me f(x) = -x + ln x. Now, to find the maximum value, I should take the derivative of f(x) with respect to x and set it equal to zero to find critical points.So, f'(x) = derivative of (-x) + derivative of ln x. The derivative of -x is -1, and the derivative of ln x is 1/x. So, f'(x) = -1 + 1/x.Setting f'(x) = 0: -1 + 1/x = 0. Solving for x, I get 1/x = 1, so x = 1.Now, I need to check if this critical point is a maximum. I can do this by using the second derivative test or analyzing the sign changes of the first derivative.Let me compute the second derivative. The first derivative is f'(x) = -1 + 1/x, so the second derivative f''(x) is the derivative of that, which is -1/x².At x = 1, f''(1) = -1/1² = -1, which is negative. Since the second derivative is negative, the function is concave down at x = 1, meaning this critical point is a local maximum.So, the maximum value of f(x) when a = -1 is f(1). Plugging x = 1 into f(x):f(1) = -1 + ln(1) = -1 + 0 = -1.Therefore, the maximum value is -1.**2. If the maximum value of f(x) in the interval (0, e] is -3, find the value of a.**Hmm, okay, so now I need to find the value of a such that the maximum of f(x) = ax + ln x on (0, e] is -3.First, let's find the critical points by taking the derivative of f(x). f'(x) = a + 1/x.Set f'(x) = 0: a + 1/x = 0 => 1/x = -a => x = -1/a.Now, this critical point x = -1/a must lie within the interval (0, e] for it to be a candidate for maximum. So, x = -1/a must satisfy 0 < -1/a ≤ e.But since x must be positive, -1/a must be positive, which implies that a is negative. So, a < 0.Now, let's consider two cases:**Case 1: The critical point x = -1/a is within (0, e].**In this case, the maximum could be at x = -1/a or at the endpoint x = e.Compute f(-1/a): f(-1/a) = a*(-1/a) + ln(-1/a) = -1 + ln(-1/a).Compute f(e): f(e) = a*e + ln(e) = a*e + 1.We are told that the maximum value is -3. So, either f(-1/a) = -3 or f(e) = -3.But let's check which one is larger. Since f(x) is increasing before x = -1/a and decreasing after, if x = -1/a is within (0, e], then f(-1/a) is the maximum.Therefore, set f(-1/a) = -3:-1 + ln(-1/a) = -3So, ln(-1/a) = -2Exponentiating both sides: -1/a = e^{-2} => a = -1/e^{-2} = -e^{2}.Now, check if x = -1/a = -1/(-e²) = 1/e². Since 1/e² is approximately 0.135, which is less than e (~2.718), so x = 1/e² is indeed within (0, e]. So, this is valid.**Case 2: The critical point x = -1/a is outside (0, e].**If x = -1/a > e, then the function f(x) would be increasing on (0, e] because f'(x) = a + 1/x. Since a is negative, but 1/x is positive and decreasing. If x = -1/a > e, then for x in (0, e], 1/x > 1/e, so a + 1/x > a + 1/e. But since a < -1/e (because x = -1/a > e implies a < -1/e), then a + 1/x could be positive or negative.Wait, actually, if x = -1/a > e, then a = -1/x < -1/e. So, a < -1/e.In this case, the derivative f'(x) = a + 1/x. At x = e, f'(e) = a + 1/e. Since a < -1/e, f'(e) = a + 1/e < 0. So, the function is decreasing at x = e, which suggests that the maximum would be at x approaching 0 or at x = e.But as x approaches 0 from the right, ln x approaches negative infinity, so f(x) = ax + ln x approaches negative infinity (since a is negative, ax approaches positive infinity, but ln x approaches negative infinity faster). So, the function tends to negative infinity as x approaches 0.Therefore, the maximum would be at x = e. But f(e) = a*e + 1. If we set this equal to -3:a*e + 1 = -3 => a*e = -4 => a = -4/e.But wait, earlier we had a = -e² ≈ -7.389, and -4/e ≈ -1.471. But in this case, we assumed that x = -1/a > e, which would require a < -1/e ≈ -0.3679. So, a = -4/e ≈ -1.471 is less than -1/e, so it's valid.But wait, if a = -4/e, then x = -1/a = e/4 ≈ 0.679, which is less than e. So, that contradicts our assumption that x = -1/a > e.Therefore, this case doesn't hold. So, the only valid solution is a = -e².Wait, let me double-check. If a = -e², then x = -1/a = 1/e² ≈ 0.135, which is indeed less than e, so the critical point is inside the interval. Therefore, the maximum is at x = 1/e², and f(1/e²) = -3.So, the value of a is -e².**3. When a = -1, determine whether the equation |f(x)| = (ln x)/x + 1/2 has real solutions.**Alright, so again, with a = -1, f(x) = -x + ln x. We already found that the maximum of f(x) is -1 at x = 1. So, f(x) ≤ -1 for all x > 0.Therefore, |f(x)| = | -x + ln x |. Since f(x) ≤ -1, |f(x)| = -f(x) = x - ln x.So, the equation becomes x - ln x = (ln x)/x + 1/2.Let me write that down:x - ln x = (ln x)/x + 1/2.Let me rearrange terms:x - ln x - (ln x)/x - 1/2 = 0.Let me define a function h(x) = x - ln x - (ln x)/x - 1/2.We need to find if h(x) = 0 has any real solutions for x > 0.Alternatively, maybe it's easier to consider the equation as:x - ln x = (ln x)/x + 1/2.Let me denote y = ln x. Then, x = e^y.Substituting into the equation:e^y - y = y e^{-y} + 1/2.Hmm, not sure if that helps. Maybe another approach.Alternatively, let's consider the function h(x) = x - ln x - (ln x)/x - 1/2.We can analyze h(x) for x > 0.First, let's check the behavior as x approaches 0+ and as x approaches infinity.As x approaches 0+:- x approaches 0.- ln x approaches -infty.- (ln x)/x approaches -infty (since ln x is negative and 1/x is positive, so overall negative infinity).Therefore, h(x) = x - ln x - (ln x)/x - 1/2 ≈ 0 - (-infty) - (-infty) - 1/2. Hmm, this is a bit messy. Let's see:- x approaches 0.- -ln x approaches +infty.- -(ln x)/x approaches +infty (since ln x is negative, so -(ln x)/x is positive, and as x approaches 0, 1/x approaches infinity, so this term goes to +infty).Therefore, h(x) ≈ 0 + infty + infty - 1/2, which is +infty.So, as x approaches 0+, h(x) approaches +infty.As x approaches infinity:- x approaches infinity.- ln x approaches infinity, but much slower than x.- (ln x)/x approaches 0.Therefore, h(x) ≈ x - ln x - 0 - 1/2. As x approaches infinity, x - ln x approaches infinity, so h(x) approaches infinity.Now, let's check h(1):h(1) = 1 - ln 1 - (ln 1)/1 - 1/2 = 1 - 0 - 0 - 1/2 = 1/2.So, h(1) = 1/2.h(e):h(e) = e - ln e - (ln e)/e - 1/2 = e - 1 - 1/e - 1/2 ≈ 2.718 - 1 - 0.368 - 0.5 ≈ 0.85.So, h(e) ≈ 0.85.Wait, but we need to find if h(x) = 0 has any solutions. Since h(x) approaches +infty as x approaches 0 and as x approaches infinity, and h(1) = 1/2, h(e) ≈ 0.85, which are both positive. Maybe h(x) is always positive?Wait, let's check h(x) at some other points.Let me try x = 1/2:h(1/2) = (1/2) - ln(1/2) - (ln(1/2))/(1/2) - 1/2.Compute each term:- (1/2) = 0.5- ln(1/2) = -ln 2 ≈ -0.693- (ln(1/2))/(1/2) = (-0.693)/0.5 ≈ -1.386So, h(1/2) = 0.5 - (-0.693) - (-1.386) - 0.5 = 0.5 + 0.693 + 1.386 - 0.5 ≈ 2.579.Positive.How about x = 2:h(2) = 2 - ln 2 - (ln 2)/2 - 1/2 ≈ 2 - 0.693 - 0.346 - 0.5 ≈ 2 - 1.539 ≈ 0.461.Still positive.Wait, maybe h(x) is always positive? Let's see.Compute h(x) = x - ln x - (ln x)/x - 1/2.Let me see if h(x) can be negative somewhere.Wait, let's try x = e^{-2} ≈ 0.135.h(e^{-2}) = e^{-2} - ln(e^{-2}) - (ln(e^{-2}))/e^{-2} - 1/2.Compute each term:- e^{-2} ≈ 0.135- ln(e^{-2}) = -2- (ln(e^{-2}))/e^{-2} = (-2)/0.135 ≈ -14.815So, h(e^{-2}) ≈ 0.135 - (-2) - (-14.815) - 0.5 ≈ 0.135 + 2 + 14.815 - 0.5 ≈ 16.45.Still positive.Wait, maybe h(x) is always positive. Let me check the derivative to see if h(x) has any minima.Compute h'(x):h(x) = x - ln x - (ln x)/x - 1/2.h'(x) = derivative of x = 1- derivative of ln x = 1/x- derivative of (ln x)/x: use quotient rule. Let me compute d/dx [ln x / x] = (1/x * x - ln x * 1)/x² = (1 - ln x)/x².- derivative of -1/2 is 0.So, h'(x) = 1 - 1/x - (1 - ln x)/x².Simplify:h'(x) = 1 - 1/x - (1 - ln x)/x².Let me write it as:h'(x) = 1 - 1/x - 1/x² + (ln x)/x².Hmm, not sure if that helps. Maybe factor terms:h'(x) = 1 - (1/x + 1/x²) + (ln x)/x².Alternatively, let's write all terms with denominator x²:h'(x) = (x²)/x² - (x)/x² - 1/x² + (ln x)/x²= [x² - x - 1 + ln x]/x².So, h'(x) = (x² - x - 1 + ln x)/x².To find critical points, set numerator equal to zero:x² - x - 1 + ln x = 0.This seems complicated to solve analytically. Maybe we can analyze the behavior.Let me define k(x) = x² - x - 1 + ln x.We need to find if k(x) = 0 has any solutions.Compute k(1):k(1) = 1 - 1 - 1 + ln 1 = -1 + 0 = -1.k(e):k(e) = e² - e - 1 + 1 ≈ 7.389 - 2.718 - 1 + 1 ≈ 4.671.So, k(1) = -1, k(e) ≈ 4.671. Since k(x) is continuous, by Intermediate Value Theorem, there is at least one root between x = 1 and x = e.Similarly, check k(0.5):k(0.5) = (0.25) - 0.5 - 1 + ln 0.5 ≈ 0.25 - 0.5 - 1 - 0.693 ≈ -2.943.k(0.5) ≈ -2.943.k(2):k(2) = 4 - 2 - 1 + ln 2 ≈ 1 + 0.693 ≈ 1.693.So, k(x) goes from -2.943 at x=0.5 to 1.693 at x=2, so another root between 0.5 and 2.Wait, but earlier we saw that h(x) is always positive. Maybe h(x) has a minimum somewhere, but since h(x) approaches +infty at both ends, and h(x) is positive at several points, perhaps h(x) never crosses zero.Wait, but if h'(x) has roots, meaning h(x) has critical points, maybe a minimum. Let me see.Suppose h(x) has a minimum somewhere. If the minimum value is above zero, then h(x) is always positive, so no solution. If the minimum is below zero, then h(x) crosses zero twice.But from our earlier computations, h(x) is positive at x=1, x=e, x=2, x=1/2, x=e^{-2}. Maybe h(x) is always positive.Alternatively, let's try to find the minimum of h(x).Since h'(x) = (x² - x - 1 + ln x)/x².Set numerator to zero: x² - x - 1 + ln x = 0.Let me denote this as k(x) = x² - x - 1 + ln x.We saw that k(1) = -1, k(e) ≈ 4.671, so there is a root between 1 and e.Similarly, k(0.5) ≈ -2.943, k(1) = -1, so another root between 0.5 and 1.Wait, no, k(0.5) is -2.943, k(1) is -1, so it's decreasing from x=0.5 to x=1? Wait, no, because k(x) is x² - x -1 + ln x.Wait, let me compute k(0.5):k(0.5) = 0.25 - 0.5 -1 + ln 0.5 ≈ 0.25 - 0.5 -1 -0.693 ≈ -1.943.Wait, earlier I thought it was -2.943, but actually it's -1.943.Wait, let me recalculate:k(0.5) = (0.5)^2 - 0.5 -1 + ln(0.5) = 0.25 - 0.5 -1 + (-0.693) ≈ 0.25 - 0.5 = -0.25; -0.25 -1 = -1.25; -1.25 -0.693 ≈ -1.943.Yes, so k(0.5) ≈ -1.943, k(1) = -1, k(2) ≈ 1.693.So, k(x) increases from x=0.5 to x=2, crossing zero somewhere between x=1 and x=2, and again between x=2 and x=e?Wait, no, k(2) is already positive, so only one root between x=1 and x=2.Wait, let me check k(1.5):k(1.5) = (2.25) - 1.5 -1 + ln(1.5) ≈ 2.25 - 2.5 + 0.405 ≈ 0.155.So, k(1.5) ≈ 0.155.So, k(1) = -1, k(1.5) ≈ 0.155. Therefore, there is a root between x=1 and x=1.5.Similarly, k(1.25):k(1.25) = (1.5625) - 1.25 -1 + ln(1.25) ≈ 1.5625 - 2.25 + 0.223 ≈ -0.4645.So, k(1.25) ≈ -0.4645.k(1.375):k(1.375) = (1.8906) - 1.375 -1 + ln(1.375) ≈ 1.8906 - 2.375 + 0.318 ≈ -0.1664.Still negative.k(1.4375):k(1.4375) ≈ (2.066) - 1.4375 -1 + ln(1.4375) ≈ 2.066 - 2.4375 + 0.364 ≈ 0.0.Approximately zero. So, the root is around x ≈ 1.4375.So, h(x) has a critical point around x ≈ 1.4375.Now, let's compute h(x) at this critical point to see if it's a minimum or maximum.Wait, since k(x) = x² - x -1 + ln x, and h'(x) = k(x)/x².So, when k(x) = 0, h'(x) = 0.To determine if it's a minimum or maximum, let's look at the sign of h'(x) around x ≈1.4375.For x < 1.4375, say x=1.4:k(1.4) = (1.96) -1.4 -1 + ln(1.4) ≈ 1.96 - 2.4 + 0.336 ≈ -0.104.So, k(1.4) < 0, so h'(x) < 0.For x >1.4375, say x=1.5:k(1.5) ≈0.155 >0, so h'(x) >0.Therefore, at x ≈1.4375, h(x) has a minimum.So, the minimum value of h(x) is at x ≈1.4375.Let me compute h(1.4375):h(1.4375) = 1.4375 - ln(1.4375) - (ln(1.4375))/1.4375 - 0.5.Compute each term:- 1.4375 ≈1.4375- ln(1.4375) ≈0.364- (ln(1.4375))/1.4375 ≈0.364 /1.4375 ≈0.253So, h(1.4375) ≈1.4375 -0.364 -0.253 -0.5 ≈1.4375 -1.117 ≈0.3205.So, the minimum value of h(x) is approximately 0.3205, which is positive.Therefore, h(x) is always positive for all x >0, meaning the equation h(x)=0 has no real solutions.Therefore, the equation |f(x)| = (ln x)/x + 1/2 has no real solutions when a = -1.

💡Alright, so I've got this vector problem here. Let me try to figure it out step by step. Okay, the problem says:"Given vectors a and b in the same plane, with a = (1,2) and |b| = (1/2)|a|, if a + 2b is perpendicular to 2a - b, what is the angle between a and b?"The options are 0, π/4, 2π/3, and π. Hmm, okay. So I need to find the angle between vectors a and b given these conditions.First, let me write down what I know. Vector a is given as (1,2). So, its magnitude |a| can be calculated using the formula for the magnitude of a vector. That's sqrt(1^2 + 2^2) = sqrt(1 + 4) = sqrt(5). Got that. So |a| is sqrt(5).Then, |b| is half of |a|, so that would be (1/2)*sqrt(5). Okay, so |b| = sqrt(5)/2.Now, the key part is that a + 2b is perpendicular to 2a - b. When two vectors are perpendicular, their dot product is zero. So, I can write that (a + 2b) · (2a - b) = 0.Let me compute this dot product. Expanding it out, it should be:a · 2a + a · (-b) + 2b · 2a + 2b · (-b) = 0Wait, that might not be the most efficient way. Maybe I should use the distributive property properly.So, (a + 2b) · (2a - b) = a · 2a + a · (-b) + 2b · 2a + 2b · (-b)Simplify each term:a · 2a = 2(a · a) = 2|a|^2a · (-b) = - (a · b)2b · 2a = 4(b · a) = 4(a · b) because dot product is commutative2b · (-b) = -2(b · b) = -2|b|^2So putting it all together:2|a|^2 - (a · b) + 4(a · b) - 2|b|^2 = 0Combine like terms:2|a|^2 + 3(a · b) - 2|b|^2 = 0Okay, so that's the equation we get from the perpendicular condition.Now, let's plug in the values we know.We already found |a| = sqrt(5), so |a|^2 = (sqrt(5))^2 = 5.Similarly, |b| = sqrt(5)/2, so |b|^2 = (sqrt(5)/2)^2 = (5)/4.So substituting these into the equation:2*5 + 3(a · b) - 2*(5/4) = 0Compute each term:2*5 = 102*(5/4) = 10/4 = 5/2So the equation becomes:10 + 3(a · b) - 5/2 = 0Combine the constants:10 - 5/2 = (20/2 - 5/2) = 15/2So now we have:15/2 + 3(a · b) = 0Solving for (a · b):3(a · b) = -15/2Divide both sides by 3:a · b = -15/6 = -5/2Okay, so the dot product of a and b is -5/2.Now, the dot product formula is also given by:a · b = |a||b|cosθWhere θ is the angle between vectors a and b.We can use this to find θ.We already know |a| = sqrt(5), |b| = sqrt(5)/2, and a · b = -5/2.So plugging into the formula:-5/2 = (sqrt(5)) * (sqrt(5)/2) * cosθSimplify the right side:(sqrt(5))*(sqrt(5)) = 5, so:-5/2 = (5/2) * cosθSo, we have:-5/2 = (5/2) cosθDivide both sides by (5/2):(-5/2) / (5/2) = cosθSimplify:-1 = cosθSo, cosθ = -1What angle has a cosine of -1? That's π radians, or 180 degrees.So, the angle between vectors a and b is π.Looking back at the options, D is π, so that should be the answer.Wait, let me double-check my steps to make sure I didn't make a mistake.1. Calculated |a| correctly: sqrt(1 + 4) = sqrt(5). That seems right.2. |b| is half of |a|, so sqrt(5)/2. Correct.3. Set up the dot product equation because the vectors are perpendicular: (a + 2b) · (2a - b) = 0. That makes sense.4. Expanded the dot product correctly: a · 2a = 2|a|^2 a · (-b) = -a · b 2b · 2a = 4a · b 2b · (-b) = -2|b|^2 So, combining: 2|a|^2 + 3a · b - 2|b|^2 = 0. That seems correct.5. Plugged in |a|^2 = 5 and |b|^2 = 5/4. So, 2*5 = 10, 2*(5/4) = 5/2. Equation becomes 10 + 3(a · b) - 5/2 = 0. 10 - 5/2 is 15/2, so 15/2 + 3(a · b) = 0. Solving for a · b: 3(a · b) = -15/2 => a · b = -5/2. That seems correct.6. Then, using the dot product formula: a · b = |a||b|cosθ Plugged in the values: -5/2 = (sqrt(5))*(sqrt(5)/2)*cosθ Simplify: (sqrt(5))*(sqrt(5)) = 5, so 5/2 cosθ = -5/2 Therefore, cosθ = -1, so θ = π. That seems correct.So, all steps check out. I think the answer is indeed π, which is option D.**Final Answer**boxed{D}

💡Okay, so I have this problem about a square EFGH on a 6 by 6 grid. The vertices are given as E(4,5), F(6,2), G(3,0), and H(1,3). I need to find the product of the area and the perimeter of this square. Hmm, let me break this down step by step.First, I remember that to find the area and perimeter of a square, I need to know the length of its sides. Since it's a square, all sides should be equal, and the diagonals should also be equal. But wait, I should verify if EFGH is indeed a square by checking the lengths of all sides and maybe the diagonals.Let me recall the distance formula between two points (x1, y1) and (x2, y2): it's sqrt[(x2 - x1)^2 + (y2 - y1)^2]. I'll use this to calculate the lengths of the sides EF, FG, GH, and HE.Starting with EF: E is (4,5) and F is (6,2). Plugging into the distance formula: sqrt[(6 - 4)^2 + (2 - 5)^2] = sqrt[(2)^2 + (-3)^2] = sqrt[4 + 9] = sqrt[13]. Okay, so EF is sqrt(13) cm.Next, FG: F is (6,2) and G is (3,0). Distance is sqrt[(3 - 6)^2 + (0 - 2)^2] = sqrt[(-3)^2 + (-2)^2] = sqrt[9 + 4] = sqrt[13]. So FG is also sqrt(13) cm.Then, GH: G is (3,0) and H is (1,3). Distance is sqrt[(1 - 3)^2 + (3 - 0)^2] = sqrt[(-2)^2 + (3)^2] = sqrt[4 + 9] = sqrt[13]. GH is sqrt(13) cm as well.Lastly, HE: H is (1,3) and E is (4,5). Distance is sqrt[(4 - 1)^2 + (5 - 3)^2] = sqrt[(3)^2 + (2)^2] = sqrt[9 + 4] = sqrt[13]. So HE is sqrt(13) cm.Alright, all sides are equal in length, which is a good sign. But just to be thorough, I should check the diagonals to confirm it's a square. In a square, the diagonals are equal and longer than the sides.Let's find the diagonals. Diagonals in a square connect opposite vertices. So, the diagonals would be EG and FH.First, EG: E is (4,5) and G is (3,0). Distance is sqrt[(3 - 4)^2 + (0 - 5)^2] = sqrt[(-1)^2 + (-5)^2] = sqrt[1 + 25] = sqrt[26].Next, FH: F is (6,2) and H is (1,3). Distance is sqrt[(1 - 6)^2 + (3 - 2)^2] = sqrt[(-5)^2 + (1)^2] = sqrt[25 + 1] = sqrt[26].So both diagonals are sqrt(26) cm, which is consistent with a square. Great, so EFGH is indeed a square with all sides equal to sqrt(13) cm.Now, moving on to calculating the area and perimeter.For a square, the area is side length squared. So, area = (sqrt(13))^2 = 13 cm².The perimeter of a square is 4 times the side length. So, perimeter = 4 * sqrt(13) cm.The problem asks for the product of the area and the perimeter. So, I need to multiply 13 cm² by 4 sqrt(13) cm.Let me compute that: 13 * 4 = 52, and then we have sqrt(13) cm. So, the product is 52 sqrt(13) cm³.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, the side lengths: all are sqrt(13), correct. Diagonals are sqrt(26), which is sqrt(2) times the side length, as expected in a square. So, that checks out.Area: (sqrt(13))² is indeed 13. Perimeter: 4 * sqrt(13) is correct. Multiplying 13 by 4 gives 52, and then we have sqrt(13). So, 52 sqrt(13) is the product.I think that's it. I don't see any errors in my calculations. So, the final answer should be 52 sqrt(13).**Final Answer**The product of the area and perimeter of square EFGH is boxed{52sqrt{13}}.

💡First, I need to determine the probability that it rains on at least one day over the weekend. The forecast provides the probabilities of rain on Saturday and Sunday, which are 30% and 60%, respectively. To find the probability that it rains on at least one day, it's often easier to first calculate the probability that it does not rain at all during the weekend and then subtract that from 1.1. **Calculate the probability that it does not rain on Saturday:** [ P(text{No rain on Saturday}) = 1 - P(text{Rain on Saturday}) = 1 - 0.30 = 0.70 ]2. **Calculate the probability that it does not rain on Sunday:** [ P(text{No rain on Sunday}) = 1 - P(text{Rain on Sunday}) = 1 - 0.60 = 0.40 ]3. **Calculate the probability that it does not rain on both days:** Since the probabilities are independent, multiply the probabilities of no rain on each day: [ P(text{No rain on both days}) = P(text{No rain on Saturday}) times P(text{No rain on Sunday}) = 0.70 times 0.40 = 0.28 ]4. **Calculate the probability that it rains on at least one day:** Subtract the probability of no rain on both days from 1: [ P(text{Rain on at least one day}) = 1 - P(text{No rain on both days}) = 1 - 0.28 = 0.72 ]Therefore, the probability that it rains over the weekend is 72%.

💡Okay, so I have this problem here: Bob traveled 80 kilometers by taxi and was charged 200. The fare is directly proportional to the distance traveled. Now, they're asking how much Bob would be charged if he traveled 120 kilometers. Hmm, alright, let's break this down step by step.First, I need to understand what "directly proportional" means. From what I remember in math class, if two things are directly proportional, it means that as one increases, the other increases at the same rate. So, in this case, the fare (which is the money Bob pays) increases as the distance he travels increases. That makes sense because the longer the ride, the more it should cost, right?So, if I think about it, there must be a constant rate or a fixed amount per kilometer that the taxi charges. Let's call this constant rate "k." So, the total fare Bob pays would be equal to this constant rate multiplied by the distance he traveled. Mathematically, that would be:Fare = k × DistanceGiven that, when Bob traveled 80 kilometers, he was charged 200. So, plugging those numbers into the equation:200 = k × 80Now, I need to find the value of "k." To do that, I can rearrange the equation to solve for "k." So, dividing both sides by 80:k = 200 / 80Let me calculate that. 200 divided by 80 is... well, 80 goes into 200 two times because 80 × 2 = 160, and that leaves a remainder of 40. Then, 80 goes into 40 half a time. So, altogether, that's 2.5. So, k = 2.5.Wait, let me double-check that. 80 × 2.5 should equal 200. 80 × 2 is 160, and 80 × 0.5 is 40. Adding those together, 160 + 40 = 200. Yep, that checks out. So, the constant rate is 2.5 per kilometer.Now, the question is asking how much Bob would be charged for traveling 120 kilometers. Using the same formula:Fare = k × DistanceWe already know that k is 2.5, and the new distance is 120 kilometers. Plugging those in:Fare = 2.5 × 120Let me calculate that. 2 × 120 is 240, and 0.5 × 120 is 60. Adding those together, 240 + 60 = 300. So, the fare for 120 kilometers should be 300.Wait a second, let me make sure I didn't make any mistakes here. If 80 kilometers cost 200, then 120 kilometers is 1.5 times longer (since 120 divided by 80 is 1.5). So, the fare should also be 1.5 times more. Let's see: 200 × 1.5 is 300. Yep, that matches what I got earlier. So, that seems consistent.Another way to think about it is setting up a proportion. If 80 kilometers cost 200, then 120 kilometers would cost "x" dollars. So, the proportion would be:80 km / 200 = 120 km / xOr, to make it clearer:80 / 200 = 120 / xWait, actually, since fare is directly proportional to distance, the ratio of fare to distance should be the same. So, it's better to set it up as:200 / 80 = x / 120That way, the fare per kilometer is consistent. So, solving for x:200 / 80 = x / 120Cross-multiplying:200 × 120 = 80 × x24,000 = 80xNow, dividing both sides by 80:x = 24,000 / 80x = 300So, again, I get 300. This method also confirms my earlier calculation.Just to be thorough, let me think about unit rates. If Bob is charged 200 for 80 kilometers, then per kilometer, it's 200 divided by 80, which is 2.5 per kilometer, as I found earlier. So, for 120 kilometers, it's 120 × 2.5, which is 300. Yep, consistent again.I guess another way to visualize this is by plotting it on a graph. If I put distance on the x-axis and fare on the y-axis, since they're directly proportional, it should be a straight line passing through the origin. The slope of this line would be the constant of proportionality, which is 2.5 per kilometer. So, starting from (0,0), for every kilometer, the fare increases by 2.5. So, at 80 kilometers, it's 200, and at 120 kilometers, it's 300. That makes sense.I could also use ratios to think about it. The ratio of the new distance to the original distance is 120/80, which simplifies to 3/2 or 1.5. So, the fare should also increase by the same ratio. Therefore, multiplying the original fare by 1.5: 200 × 1.5 = 300. Yep, same answer.Is there any chance I could have messed up the calculations? Let me check once more. 80 kilometers cost 200, so per kilometer, it's 2.5. 120 kilometers would then be 120 × 2.5. 100 kilometers would be 100 × 2.5 = 250, and 20 kilometers would be 20 × 2.5 = 50. Adding those together, 250 + 50 = 300. Yep, that's correct.Alternatively, thinking about it in fractions, 2.5 is the same as 5/2 per kilometer. So, for 120 kilometers, it's 120 × (5/2). 120 divided by 2 is 60, and 60 × 5 is 300. So, again, 300.I think I've approached this problem from multiple angles: using the constant of proportionality, setting up proportions, calculating unit rates, using ratios, and even visualizing it on a graph. All these methods consistently lead me to the conclusion that the fare for 120 kilometers would be 300. I don't see any discrepancies or errors in my calculations, so I feel confident that 300 is the correct answer.