🧠:<think>Okay, so Ms. Johnson is a retired theater teacher trying stand-up comedy, and she's got some math problems to solve for her performance. Let me try to figure these out step by step.Starting with the first problem: She has 15 skits, each with a setup time that's normally distributed with a mean of 3 minutes and a standard deviation of 0.5 minutes. Each punchline is exactly 1 minute. She wants to perform 8 skits, and the total expected performance time should be exactly 32 minutes. We need to find how many different combinations of 8 skits she can choose.Hmm. So, each skit has a setup time that's a random variable, but the punchline is fixed at 1 minute. So, for each skit, the total time is setup time + punchline time. The setup time is normally distributed, but since we're dealing with expected value, maybe we can just use the mean for each skit's setup time?Wait, the problem says the total expected performance time should be exactly 32 minutes. So, expectation is linear, right? So, the expected total setup time plus the expected total punchline time should equal 32.Each punchline is 1 minute, so for 8 skits, that's 8 minutes. Therefore, the expected total setup time should be 32 - 8 = 24 minutes. Since each setup has an expected value of 3 minutes, 8 skits would have an expected setup time of 8 * 3 = 24 minutes. So, actually, every combination of 8 skits would have the same expected total performance time of 32 minutes, because expectation is linear and doesn't depend on the specific skits chosen, just the number.Wait, does that mean that all combinations are acceptable? So, the number of different combinations is just the number of ways to choose 8 skits out of 15, which is the combination C(15,8).Let me verify that. The expected setup time per skit is 3 minutes, so 8 skits would have 24 minutes expected setup. Punchlines are 8 minutes, so total expected time is 32 minutes. Therefore, regardless of which skits she picks, the expected total time is the same. So, the number of combinations is just the number of ways to choose 8 skits from 15.Calculating that, C(15,8) is equal to 6435. So, the answer is 6435 different combinations.Wait, but hold on. The setup times are normally distributed, but does that affect the expectation? No, because expectation is linear regardless of the distribution. So, even though each setup time is a random variable, the expected total setup time is still 8 * 3 = 24 minutes. So, yes, all combinations are acceptable. Therefore, the number is just the combination.So, for the first problem, the answer is 6435.Moving on to the second problem: Ms. Johnson is engaging the audience with interactive questions, and she models the audience's response time as a Poisson process with an average rate of 2 responses per minute. We need to find the probability that she will receive exactly 5 responses in a 3-minute interval.Alright, Poisson processes. The number of events in a given interval follows a Poisson distribution. The formula for the Poisson probability is P(k) = (λ^k * e^{-λ}) / k!, where λ is the expected number of events in the interval.Here, the rate is 2 responses per minute, so over 3 minutes, the expected number λ is 2 * 3 = 6.So, we need to find P(5) = (6^5 * e^{-6}) / 5!.Calculating that:First, 6^5 is 6*6*6*6*6 = 7776.Then, e^{-6} is approximately 0.002478752.5! is 120.So, putting it all together: (7776 * 0.002478752) / 120.First, multiply 7776 by 0.002478752:7776 * 0.002478752 ≈ Let's compute that.7776 * 0.002 = 15.5527776 * 0.000478752 ≈ 7776 * 0.0004 = 3.1104; 7776 * 0.000078752 ≈ approximately 0.611.So, total ≈ 15.552 + 3.1104 + 0.611 ≈ 19.2734.Then, divide by 120: 19.2734 / 120 ≈ 0.1606.So, approximately 0.1606, or 16.06%.But let me compute it more accurately.Compute 6^5 = 7776.Compute e^{-6} ≈ 0.002478752.Multiply 7776 * 0.002478752:Let me do this step by step.7776 * 0.002 = 15.5527776 * 0.0004 = 3.11047776 * 0.000078752 ≈ Let's compute 7776 * 0.00007 = 0.54432; 7776 * 0.000008752 ≈ approximately 0.068.So, total ≈ 15.552 + 3.1104 + 0.54432 + 0.068 ≈ 19.27472.Then, 19.27472 / 120 ≈ 0.1606226667.So, approximately 0.1606, which is about 16.06%.Alternatively, using a calculator for more precision:Compute 7776 * 0.002478752:7776 * 0.002478752 = 7776 * (2478.752 / 1,000,000) ≈ 7776 * 0.002478752.Let me compute 7776 * 0.002478752:First, 7776 * 0.002 = 15.5527776 * 0.0004 = 3.11047776 * 0.000078752 ≈ 7776 * 0.00007 = 0.54432; 7776 * 0.000008752 ≈ 0.068.Adding up: 15.552 + 3.1104 = 18.6624; 18.6624 + 0.54432 = 19.20672; 19.20672 + 0.068 ≈ 19.27472.Divide by 120: 19.27472 / 120 ≈ 0.1606226667.So, approximately 0.1606, which is 16.06%.Alternatively, using a calculator, 6^5 is 7776, e^{-6} is approximately 0.002478752, so 7776 * 0.002478752 ≈ 19.27472, divided by 120 is approximately 0.1606.So, the probability is approximately 0.1606, or 16.06%.But maybe we can write it more precisely.Alternatively, using the exact formula:P(5) = (6^5 * e^{-6}) / 5! = (7776 * e^{-6}) / 120.We can compute this as:7776 / 120 = 64.8So, 64.8 * e^{-6} ≈ 64.8 * 0.002478752 ≈ Let's compute 64.8 * 0.002 = 0.1296; 64.8 * 0.000478752 ≈ approximately 0.0309.Adding up: 0.1296 + 0.0309 ≈ 0.1605.So, same result. So, approximately 0.1605 or 16.05%.Therefore, the probability is approximately 0.1606, which is about 16.06%.So, rounding to four decimal places, it's approximately 0.1606.Alternatively, if we use more precise calculation:Compute 6^5 = 7776.Compute e^{-6} ≈ 0.002478752176666358.Multiply 7776 * 0.002478752176666358:Let me compute 7776 * 0.002478752176666358.First, 7776 * 0.002 = 15.5527776 * 0.000478752176666358 ≈ Let's compute 7776 * 0.0004 = 3.1104; 7776 * 0.000078752176666358 ≈ approximately 0.611.So, total ≈ 15.552 + 3.1104 + 0.611 ≈ 19.2734.Divide by 120: 19.2734 / 120 ≈ 0.1606116667.So, approximately 0.1606.Therefore, the probability is approximately 0.1606, or 16.06%.So, to summarize:1. The number of combinations is C(15,8) = 6435.2. The probability is approximately 0.1606.Final Answer1. The number of different combinations is boxed{6435}.2. The probability is approximately boxed{0.1606}.</think>

🧠:<think>Okay, so I have these two math problems to solve, and I need to figure them out step by step. Let me start with the first one about the Parthenon and the Golden Ratio.1. The problem says that the height of the Parthenon is approximately 13.72 meters, and the ratio of its height to its width is supposed to be the Golden Ratio, which is about 1.618. I need to find the width, W, to the nearest hundredth of a meter.Hmm, okay. So, the Golden Ratio is the ratio of height to width. That means height divided by width equals the Golden Ratio. So, mathematically, that would be:Height / Width = Golden RatioPlugging in the numbers:13.72 / W = 1.618I need to solve for W. So, I can rearrange the equation:W = 13.72 / 1.618Let me calculate that. I can use a calculator for this.First, 13.72 divided by 1.618. Let me do that division.So, 13.72 ÷ 1.618 ≈ ?Let me compute this step by step.1.618 goes into 13.72 how many times?Well, 1.618 x 8 = 12.944Subtracting that from 13.72: 13.72 - 12.944 = 0.776So, 0.776 is the remainder. Now, bring down a zero to make it 7.76.1.618 goes into 7.76 approximately 4 times because 1.618 x 4 = 6.472Subtracting: 7.76 - 6.472 = 1.288Bring down another zero to make it 12.88.1.618 goes into 12.88 about 8 times because 1.618 x 8 = 12.944Wait, that's actually a bit more than 12.88. Let me check:1.618 x 7 = 11.326Subtracting: 12.88 - 11.326 = 1.554So, it's about 7 times with some remainder.Wait, maybe I should just do this division more accurately.Alternatively, I can use the calculator function on my computer or phone. Let me just compute 13.72 / 1.618.Calculating 13.72 ÷ 1.618:1.618 × 8 = 12.94413.72 - 12.944 = 0.776So, 0.776 / 1.618 ≈ 0.479So, total is 8.479So, approximately 8.48 meters.Wait, let me confirm with a calculator:13.72 ÷ 1.618 ≈ 8.48Yes, that seems right.So, the width is approximately 8.48 meters.Wait, but let me double-check my division because sometimes I make mistakes.Alternatively, I can use cross-multiplication.If 13.72 / W = 1.618, then W = 13.72 / 1.618.Calculating 13.72 divided by 1.618:Let me write it as 13.72 ÷ 1.618.Convert 1.618 to a fraction: approximately 1618/1000.So, 13.72 ÷ (1618/1000) = 13.72 × (1000/1618) = (13.72 × 1000) / 1618 = 13720 / 1618 ≈ ?Calculating 13720 ÷ 1618:1618 × 8 = 12944Subtract from 13720: 13720 - 12944 = 776Bring down a zero: 77601618 × 4 = 6472Subtract: 7760 - 6472 = 1288Bring down another zero: 128801618 × 8 = 12944Wait, that's too much. So, 1618 × 7 = 11326Subtract: 12880 - 11326 = 1554So, so far, we have 8.47...Wait, so 8.47 with some remainder.So, approximately 8.479, which rounds to 8.48.Yes, so W ≈ 8.48 meters.Okay, that seems solid.2. The second problem is about the Fibonacci sequence and the Golden Ratio. The professional has 45 minutes to solve a math problem and knows the first 10 Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.They need to use these numbers to approximate the Golden Ratio by taking the ratio of the 10th Fibonacci number to the 9th. Then, analyze the error in this approximation compared to the actual Golden Ratio, expressed as a percentage difference.First, let's recall that the Fibonacci sequence is defined such that each number is the sum of the two preceding ones. The sequence given is correct: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.So, the 10th Fibonacci number is 55, and the 9th is 34.Therefore, the ratio is 55 / 34.Calculating that:55 ÷ 34 ≈ 1.6176Wait, let me compute it precisely.34 goes into 55 once, with a remainder of 21.So, 1.0, and then 21/34.21 ÷ 34 ≈ 0.6176So, total is approximately 1.6176.The actual Golden Ratio is approximately 1.61803398875.So, the approximation using the 10th and 9th Fibonacci numbers is about 1.6176.Now, to find the percentage error.Percentage error is calculated as:|Approximate Value - Actual Value| / Actual Value × 100%So, plugging in the numbers:|1.6176 - 1.61803398875| / 1.61803398875 × 100%First, compute the absolute difference:1.61803398875 - 1.6176 = 0.00043398875So, the absolute difference is approximately 0.000434.Now, divide that by the actual value:0.000434 / 1.61803398875 ≈ ?Calculating that:0.000434 ÷ 1.61803398875 ≈ 0.000268Multiply by 100% to get the percentage:0.000268 × 100 ≈ 0.0268%So, approximately 0.0268% error.Rounding to a reasonable decimal place, maybe 0.027%.But let me double-check the calculations to ensure accuracy.First, 55 / 34:34 × 1.6176 = 34 × 1 + 34 × 0.617634 × 1 = 3434 × 0.6176: Let's compute 34 × 0.6 = 20.4, 34 × 0.0176 ≈ 0.5984So, total ≈ 20.4 + 0.5984 ≈ 20.9984So, 34 × 1.6176 ≈ 34 + 20.9984 ≈ 54.9984, which is approximately 55. So, that's correct.The actual Golden Ratio is approximately 1.61803398875.So, the approximate ratio is 1.6176, which is very close.The difference is 1.61803398875 - 1.6176 = 0.00043398875.Divide that by the actual value:0.00043398875 / 1.61803398875 ≈ 0.000268Multiply by 100: 0.0268%, which is approximately 0.027%.So, the percentage error is about 0.027%.Alternatively, if we want more decimal places, it's approximately 0.0268%, which is roughly 0.027%.So, that's the error.Wait, let me confirm the division:0.00043398875 ÷ 1.61803398875.Let me write it as 4.3398875 × 10^-4 divided by 1.61803398875.So, 4.3398875 / 1618.03398875 ≈ ?Wait, no, 4.3398875 × 10^-4 divided by 1.61803398875 is equal to (4.3398875 / 1.61803398875) × 10^-4.Compute 4.3398875 / 1.61803398875:1.61803398875 × 2.68 ≈ 4.3398875Yes, because 1.61803398875 × 2 = 3.23606797751.61803398875 × 0.68 ≈ 1.61803398875 × 0.6 = 0.970820393251.61803398875 × 0.08 ≈ 0.1294427191So, total ≈ 0.97082039325 + 0.1294427191 ≈ 1.10026311235So, 3.2360679775 + 1.10026311235 ≈ 4.33633108985Which is very close to 4.3398875.So, the multiplier is approximately 2.68.Therefore, 4.3398875 / 1.61803398875 ≈ 2.68Therefore, 2.68 × 10^-4 = 0.000268Multiply by 100%: 0.0268%So, yes, that's correct.Therefore, the percentage error is approximately 0.0268%, which is about 0.027%.So, summarizing:1. The width of the Parthenon is approximately 8.48 meters.2. The ratio of the 10th Fibonacci number to the 9th is approximately 1.6176, which is about 0.027% less than the actual Golden Ratio.I think that's all. Let me just make sure I didn't make any calculation errors.For the first problem, 13.72 divided by 1.618 is indeed approximately 8.48. I think that's correct.For the second problem, 55/34 is approximately 1.6176, and the error is about 0.027%. That seems right.Yeah, I think I'm confident with these answers.Final Answer1. The width of the Parthenon is boxed{8.48} meters.2. The ratio is approximately boxed{1.6176}, with an error of boxed{0.027%}.</think>

🧠:<think>Okay, so I have this problem about an ambitious literary agent trying to maximize the combined productivity of 10 authors over a 5-year period. Each author's productivity is modeled by an exponential function, P_i(t) = a_i * e^{b_i t}. I need to figure out two things: first, the total productivity function T(t) over the 5 years, and second, the conditions on a_i and b_i such that T(t) is maximized at t=5.Let me start with the first part. The total productivity function T(t) should be the sum of each individual author's productivity over time. Since each author's productivity is given by P_i(t), I think T(t) would just be the sum from i=1 to 10 of P_i(t). So, mathematically, that would be:T(t) = Σ_{i=1}^{10} a_i * e^{b_i t}That seems straightforward. Each author contributes their own exponential function, and the total is just the sum of all these. So, for any time t between 0 and 5, T(t) gives the total productivity.Now, moving on to the second part. The agent wants to maximize T(t) at t=5. So, we need to ensure that t=5 is the point where T(t) reaches its maximum value over the interval [0,5].To find the maximum, I remember that we can use calculus. Specifically, we can take the derivative of T(t) with respect to t, set it equal to zero, and solve for t. The solution should be t=5 for it to be the maximum.So, let's compute the derivative of T(t). Since T(t) is the sum of exponentials, the derivative T’(t) will be the sum of the derivatives of each P_i(t). The derivative of a_i * e^{b_i t} with respect to t is a_i * b_i * e^{b_i t}. Therefore:T’(t) = Σ_{i=1}^{10} a_i * b_i * e^{b_i t}To find critical points, set T’(t) = 0:Σ_{i=1}^{10} a_i * b_i * e^{b_i t} = 0Hmm, but e^{b_i t} is always positive for any real b_i and t. So, each term in the sum is a_i * b_i times a positive number. Therefore, the sum can only be zero if each term is zero or if the positive and negative terms cancel out.But wait, a_i and b_i are constants specific to each author. I don't know if they can be positive or negative. Productivity is likely a positive quantity, so a_i should be positive. As for b_i, if b_i is positive, the productivity grows exponentially; if b_i is negative, it decays. But since we're talking about productivity over time, it's possible that some authors have increasing productivity (b_i positive) and others have decreasing (b_i negative). But to maximize T(t) at t=5, we need to ensure that t=5 is a maximum. So, the derivative at t=5 should be zero, and the second derivative should be negative (to confirm it's a maximum).So, first condition: T’(5) = 0That is:Σ_{i=1}^{10} a_i * b_i * e^{b_i * 5} = 0Second condition: T''(t) < 0 at t=5Compute T''(t):T''(t) = Σ_{i=1}^{10} a_i * b_i^2 * e^{b_i t}Since e^{b_i t} is always positive, and a_i is positive (as productivity can't be negative), the sign of each term in T''(t) depends on b_i^2. But b_i^2 is always non-negative, so each term in T''(t) is non-negative. Therefore, T''(t) is the sum of non-negative terms, which means T''(t) is non-negative. Wait, that's a problem. If T''(t) is non-negative everywhere, then any critical point would be a minimum, not a maximum. But we need t=5 to be a maximum. This seems contradictory.Hmm, maybe I made a mistake. Let me think again.If all the terms in T''(t) are non-negative, then T''(t) is non-negative everywhere. So, the function T(t) is convex, meaning any critical point is a minimum. Therefore, to have a maximum at t=5, the function must be decreasing on [0,5], which would mean that the maximum is at t=0. But that's not what we want.Alternatively, if T(t) is increasing on [0,5], then the maximum would be at t=5. But in that case, the derivative T’(t) would be positive for all t in [0,5], meaning T’(5) > 0, which contradicts the condition for a maximum.Wait, so maybe the function can't have a maximum at t=5 if all the second derivatives are non-negative. Unless some of the b_i are negative, making some terms in T’(t) negative.But even so, if some b_i are positive and some are negative, the derivative T’(t) could have both positive and negative contributions. So, maybe at t=5, the positive and negative contributions cancel out, making T’(5)=0, which would be a critical point. But since T''(t) is still the sum of non-negative terms, it would be non-negative, meaning that critical point would be a minimum, not a maximum.This seems like a contradiction. How can we have a maximum at t=5 if the second derivative is non-negative?Wait, perhaps I need to reconsider. Maybe the function T(t) is not necessarily convex everywhere. If some b_i are positive and some are negative, the second derivative could be positive or negative depending on the combination. But no, because each term in T''(t) is a_i * b_i^2 * e^{b_i t}, which is always non-negative since a_i > 0 and b_i^2 >=0. So, T''(t) is always non-negative, meaning T(t) is convex. Therefore, any critical point is a minimum, not a maximum.So, if T(t) is convex, the maximum on the interval [0,5] must occur at one of the endpoints, either t=0 or t=5. Therefore, to have the maximum at t=5, we need T(5) >= T(0).But T(0) is the sum of a_i * e^{0} = sum of a_i. T(5) is the sum of a_i * e^{5 b_i}. So, to have T(5) >= T(0), we need:Σ_{i=1}^{10} a_i * e^{5 b_i} >= Σ_{i=1}^{10} a_iWhich simplifies to:Σ_{i=1}^{10} a_i (e^{5 b_i} - 1) >= 0So, the condition is that the sum of a_i (e^{5 b_i} - 1) is non-negative.But wait, if all b_i are positive, then e^{5 b_i} >1, so each term is positive, making the sum positive. Therefore, T(5) > T(0), so the maximum is at t=5.Alternatively, if some b_i are negative, then e^{5 b_i} <1, making those terms negative. So, the overall sum could be positive or negative depending on the balance between positive and negative contributions.But the problem is that even if T(5) is greater than T(0), the function T(t) is convex, so it might have a minimum somewhere in between, but the maximum would still be at t=5 if T(5) > T(0). Because in a convex function on an interval, the maximum is at one of the endpoints.Wait, yes, that's correct. For a convex function on [a,b], the maximum occurs at one of the endpoints. So, if T(5) > T(0), then the maximum is at t=5. If T(5) < T(0), the maximum is at t=0. If they are equal, then it's constant.Therefore, to have the maximum at t=5, we need T(5) >= T(0). So, the condition is:Σ_{i=1}^{10} a_i (e^{5 b_i} - 1) >= 0But the problem asks for conditions on a_i and b_i such that T(t) is maximized at t=5. So, that would be that the sum above is non-negative.Alternatively, if we think about the derivative, since T(t) is convex, the derivative at t=5 must be non-negative, and the derivative at t=0 must be non-positive. Because if the function is increasing at t=5, it's still going up, so the maximum would be beyond t=5, but since our interval is up to t=5, the maximum is at t=5.Wait, but in our case, the interval is fixed at 5 years. So, regardless of the derivative beyond t=5, within [0,5], the maximum is at t=5 if T(5) >= T(0).But let's think again. If T(t) is convex, then it curves upward. So, if T(5) > T(0), then the function is increasing over the interval, so the maximum is at t=5. If T(5) < T(0), the function is decreasing, so the maximum is at t=0. If T(5)=T(0), the function is flat, so maximum is everywhere.Therefore, the condition is simply that T(5) >= T(0), which translates to:Σ_{i=1}^{10} a_i (e^{5 b_i} - 1) >= 0So, that's the condition.But wait, the problem says "determine the conditions on the constants a_i and b_i such that the total productivity T(t) is maximized at t=5."So, the condition is that the sum of a_i (e^{5 b_i} -1) is non-negative.Alternatively, we can write it as:Σ_{i=1}^{10} a_i e^{5 b_i} >= Σ_{i=1}^{10} a_iWhich is the same as:Σ_{i=1}^{10} a_i (e^{5 b_i} - 1) >= 0So, that's the condition.But let me think if there's another way to phrase this. Maybe in terms of the growth rates. If the weighted sum of the growth factors e^{5 b_i} is at least 1, then T(5) >= T(0). So, the average growth factor, weighted by a_i, must be at least 1.Alternatively, if we consider that each author's productivity at t=5 is a_i e^{5 b_i}, and at t=0 is a_i. So, the total productivity at t=5 is the sum of a_i e^{5 b_i}, and at t=0 it's the sum of a_i. So, for the total to be higher at t=5, the sum of the growth factors must be at least 1.Therefore, the condition is that the sum of a_i (e^{5 b_i} -1) is non-negative.Alternatively, we can write it as:Σ_{i=1}^{10} a_i e^{5 b_i} >= Σ_{i=1}^{10} a_iWhich is the same as:Σ_{i=1}^{10} a_i (e^{5 b_i} - 1) >= 0So, that's the condition.But wait, is there a way to relate this to the derivative? Earlier, I thought about setting T’(5)=0, but that led to a contradiction because T''(t) is non-negative, implying a minimum. But since T(t) is convex, the maximum is at the endpoints, so setting T’(5)=0 isn't necessary unless we want a critical point at t=5, but that would be a minimum, not a maximum.Therefore, the correct condition is simply that T(5) >= T(0), which is the sum condition above.So, to summarize:1. The total productivity function is T(t) = Σ_{i=1}^{10} a_i e^{b_i t}2. The condition for T(t) to be maximized at t=5 is that Σ_{i=1}^{10} a_i (e^{5 b_i} - 1) >= 0Alternatively, written as Σ_{i=1}^{10} a_i e^{5 b_i} >= Σ_{i=1}^{10} a_iSo, that's the answer.</think>

🧠:<think>Alright, so I have this problem about a water polo player's motion in a pool, modeled by parametric equations. There are two parts: first, finding the speed at t=2 seconds, and second, determining when the trajectory makes a 45-degree angle with the x-axis. Let me tackle them one by one.Starting with part 1: Calculate the player's speed at t=2 seconds. Hmm, okay, so speed in the context of parametric equations is the magnitude of the velocity vector. Velocity is the derivative of the position vector, right? So, I need to find the derivatives of x(t) and y(t) with respect to time t, which will give me the velocity components in the x and y directions. Then, I can find the magnitude of that velocity vector at t=2.Let me write down the given parametric equations:x(t) = 3t² - 4t + 1y(t) = 2t³ - 3t² + tSo, first, I need to find dx/dt and dy/dt.Calculating dx/dt:The derivative of x(t) with respect to t is:dx/dt = d/dt [3t² - 4t + 1] = 6t - 4Similarly, the derivative of y(t) with respect to t is:dy/dt = d/dt [2t³ - 3t² + t] = 6t² - 6t + 1Okay, so velocity vector v(t) is (dx/dt, dy/dt) = (6t - 4, 6t² - 6t + 1)Now, to find the speed at t=2, I need to evaluate dx/dt and dy/dt at t=2, then compute the magnitude.Let me compute dx/dt at t=2:dx/dt at t=2: 6*(2) - 4 = 12 - 4 = 8 m/sSimilarly, dy/dt at t=2: 6*(2)² - 6*(2) + 1 = 6*4 - 12 + 1 = 24 - 12 + 1 = 13 m/sSo, the velocity vector at t=2 is (8, 13). Now, the speed is the magnitude of this vector, which is sqrt(8² + 13²).Calculating that:8² = 6413² = 16964 + 169 = 233So, speed = sqrt(233) m/sLet me compute sqrt(233). Hmm, 15² is 225, 16² is 256, so sqrt(233) is between 15 and 16. Let me see, 15.26² is approximately 233 because 15.26*15.26 ≈ 233. So, approximately 15.26 m/s. But since the problem doesn't specify rounding, I can just leave it as sqrt(233) m/s.Wait, let me double-check my derivatives. For x(t), derivative is 6t - 4, correct. For y(t), derivative is 6t² - 6t + 1, yes, that's correct. Plugging t=2 into both, 6*2 -4 is 8, 6*4 -12 +1 is 24 -12 +1 is 13. So, velocity components are correct. Then, speed is sqrt(8² +13²)=sqrt(64+169)=sqrt(233). Yep, that seems right.So, part 1 is done. The speed at t=2 is sqrt(233) m/s.Moving on to part 2: The player needs to intercept a ball at the point where their trajectory makes a 45-degree angle with the x-axis. I need to find the time t when this occurs.Alright, so the trajectory's slope is equal to the tangent of 45 degrees. Since tan(45°) is 1, the slope dy/dx should be 1.But how do I find dy/dx? Well, for parametric equations, dy/dx is (dy/dt)/(dx/dt). So, I can set (dy/dt)/(dx/dt) equal to 1 and solve for t.So, let's write that equation:(dy/dt)/(dx/dt) = 1Which means:(dy/dt) = (dx/dt)So, substituting the derivatives:6t² - 6t + 1 = 6t - 4Let me write that equation:6t² - 6t + 1 = 6t - 4Bring all terms to one side:6t² - 6t + 1 - 6t + 4 = 0Simplify:6t² - 12t + 5 = 0So, quadratic equation: 6t² -12t +5 =0Let me solve this quadratic for t.Quadratic formula: t = [12 ± sqrt(144 - 4*6*5)] / (2*6)Compute discriminant:144 - 120 = 24So, sqrt(24) = 2*sqrt(6)Thus, t = [12 ± 2sqrt(6)] / 12Simplify:t = [12 ± 2sqrt(6)] /12 = [6 ± sqrt(6)] /6 = 1 ± (sqrt(6)/6)So, t = 1 + sqrt(6)/6 or t = 1 - sqrt(6)/6Compute sqrt(6): approximately 2.449So, sqrt(6)/6 ≈ 0.408Thus, t ≈ 1 + 0.408 ≈ 1.408 seconds or t ≈ 1 - 0.408 ≈ 0.592 secondsNow, since time cannot be negative, both solutions are positive, so both are possible. But we need to check if these times are valid in the context.Wait, but let me think: the parametric equations are defined for t ≥0, so both t≈0.592 and t≈1.408 are valid. So, the player's trajectory crosses the 45-degree angle twice? Hmm, that seems possible.But let me verify if both these times actually result in the slope being 1.Alternatively, maybe I made a mistake in the algebra. Let me double-check.Starting from:(dy/dt)/(dx/dt) = 1So, (6t² -6t +1)/(6t -4) =1Cross-multiplying:6t² -6t +1 = 6t -4Bring all terms to left:6t² -6t +1 -6t +4 =0Simplify:6t² -12t +5=0Yes, that's correct. So, quadratic equation is correct.So, solutions are t = [12 ± sqrt(144 -120)] /12 = [12 ± sqrt(24)] /12 = [12 ± 2*sqrt(6)] /12 = [6 ± sqrt(6)] /6 = 1 ± sqrt(6)/6So, approximately 1.408 and 0.592. So, both are valid times.But wait, let me check if at t=0.592, dx/dt is positive or negative.Compute dx/dt at t=0.592:dx/dt =6t -4At t≈0.592, 6*0.592≈3.552, so 3.552 -4≈-0.448 m/sSo, negative velocity in x-direction. So, the player is moving left at that time.Similarly, at t≈1.408, dx/dt=6*1.408 -4≈8.448 -4≈4.448 m/s, positive.So, the slope is 1 in both cases, but in one case, the player is moving to the left (negative x velocity) and in the other, moving to the right (positive x velocity). So, both are valid points where the trajectory makes a 45-degree angle with the x-axis.But the problem says "the point where their trajectory makes a 45-degree angle with the x-axis." It doesn't specify direction, so both times are valid. However, maybe the question expects the first time it happens, which is t≈0.592 seconds, or perhaps both times.Wait, let me check the exact values.t = 1 ± sqrt(6)/6sqrt(6)≈2.449, so sqrt(6)/6≈0.408Thus, t≈1.408 and t≈0.592So, both are correct. But the problem says "the time t at which this occurs." It doesn't specify which one, so perhaps both are acceptable. But maybe in the context, the player is moving forward, so maybe t≈1.408 is the intended answer. Hmm.Alternatively, perhaps the problem expects both times. Let me see.Wait, let me think about the parametric equations. Let me see what the path looks like.x(t) = 3t² -4t +1y(t) =2t³ -3t² +tSo, x(t) is a quadratic function, opening upwards, so it's a parabola in x vs t.y(t) is a cubic function, so it's more complex.But the trajectory is in the plane, so it's a curve. The slope dy/dx is 1 at two points, as we found.But perhaps the question expects both times? Or maybe just the first time?Wait, the problem says "the point where their trajectory makes a 45-degree angle with the x-axis." It's singular, "the point," but actually, there are two such points. So, maybe both times are acceptable.But perhaps, in the context of the problem, the player is swimming from t=0 onwards, so both times are valid. So, maybe both t=1 + sqrt(6)/6 and t=1 - sqrt(6)/6 are solutions.But sqrt(6)/6 is approximately 0.408, so 1 - 0.408≈0.592, which is positive, so both are acceptable.Therefore, the times are t=1 ± sqrt(6)/6 seconds.Wait, but let me express sqrt(6)/6 as sqrt(6)/6, which can be written as (sqrt(6))/6 or simplified as (sqrt(6)/6). Alternatively, rationalizing, but I think it's fine as is.So, the exact times are t=1 ± sqrt(6)/6.But let me check if the derivatives are correct.From x(t)=3t² -4t +1, dx/dt=6t -4, correct.From y(t)=2t³ -3t² +t, dy/dt=6t² -6t +1, correct.Then, setting dy/dx=1, which is (6t² -6t +1)/(6t -4)=1So, 6t² -6t +1 =6t -4Bring all terms to left: 6t² -12t +5=0, correct.Solutions t=(12 ±sqrt(144-120))/12=(12±sqrt(24))/12=(12±2sqrt(6))/12=1±sqrt(6)/6, correct.So, yes, the solutions are correct.Therefore, the times are t=1 + sqrt(6)/6 and t=1 - sqrt(6)/6.But let me compute sqrt(6)/6:sqrt(6)=2.449, so sqrt(6)/6≈0.408, so t≈1.408 and t≈0.592.So, both are valid.But the problem says "the time t at which this occurs." It's a bit ambiguous whether it's asking for all times or just one. Since it's a 45-degree angle, which can occur at two points, I think both are correct.But maybe in the context, the player is moving forward, so t=1.408 is the time when moving in the positive x-direction, while t=0.592 is when moving in the negative x-direction. So, depending on the context, maybe both are acceptable.But since the problem doesn't specify, I think both times are correct. So, I should present both solutions.Alternatively, maybe the problem expects the answer in exact form, so t=1 ± sqrt(6)/6.But let me see if the problem expects a single answer. Hmm, the wording is "the time t at which this occurs." It's possible that both times are acceptable, so I should include both.Therefore, the times are t=1 + sqrt(6)/6 and t=1 - sqrt(6)/6 seconds.Wait, but let me check if at t=1 - sqrt(6)/6, the denominator dx/dt is not zero.At t=1 - sqrt(6)/6, dx/dt=6t -4=6*(1 - sqrt(6)/6) -4=6 - sqrt(6) -4=2 - sqrt(6)Compute 2 - sqrt(6): sqrt(6)≈2.449, so 2 -2.449≈-0.449, which is not zero, so it's valid.Similarly, at t=1 + sqrt(6)/6, dx/dt=6*(1 + sqrt(6)/6) -4=6 + sqrt(6) -4=2 + sqrt(6)≈4.449, which is also not zero.So, both times are valid, and the slope is defined.Therefore, the times are t=1 ± sqrt(6)/6 seconds.So, summarizing:1. Speed at t=2 is sqrt(233) m/s.2. The times when the trajectory makes a 45-degree angle with the x-axis are t=1 + sqrt(6)/6 and t=1 - sqrt(6)/6 seconds.But let me write sqrt(6)/6 as (sqrt(6))/6 for clarity.Alternatively, sqrt(6)/6 can be written as (sqrt(6))/6, which is approximately 0.408.So, the times are approximately 1.408 and 0.592 seconds.But since the problem didn't specify to approximate, I should present the exact values.Therefore, the times are t=1 ± sqrt(6)/6 seconds.Wait, but let me check if the problem expects a single answer or multiple. Since it's a 45-degree angle, which can occur at two points, I think both are correct.So, I think I'm done.Final Answer1. The player's speed at ( t = 2 ) seconds is (boxed{sqrt{233}}) meters per second.2. The times at which the trajectory makes a (45^circ) angle with the x-axis are (boxed{1 + dfrac{sqrt{6}}{6}}) seconds and (boxed{1 - dfrac{sqrt{6}}{6}}) seconds.</think>