🧠:<think>Alright, so I have this problem about Emily and her educational fund. It's split into two parts, and I need to figure out both. Let me start with the first one.Problem 1:They gave me that the effectiveness ( b_k = ln(k+1) ) and the grant ( a_k ) is an arithmetic sequence with ( a_1 = 500 ) and common difference ( d = 100 ). I need to find a closed-form expression for the total fund ( F(n) = sum_{k=1}^{n} (a_k cdot e^{b_k}) ).First, let me write down what I know.An arithmetic sequence has the form ( a_k = a_1 + (k - 1)d ). So plugging in the values, ( a_k = 500 + (k - 1) times 100 ). Simplifying that, ( a_k = 500 + 100k - 100 = 400 + 100k ). So, ( a_k = 100k + 400 ).Next, ( b_k = ln(k + 1) ). So, ( e^{b_k} = e^{ln(k + 1)} = k + 1 ). That simplifies things because the exponential of a natural log is just the argument itself.So, substituting back into ( F(n) ), we have:( F(n) = sum_{k=1}^{n} a_k cdot e^{b_k} = sum_{k=1}^{n} (100k + 400)(k + 1) ).Now, let me expand this expression inside the summation:( (100k + 400)(k + 1) = 100k(k + 1) + 400(k + 1) ).Breaking it down:First term: ( 100k(k + 1) = 100(k^2 + k) ).Second term: ( 400(k + 1) = 400k + 400 ).So, combining both terms:( 100k^2 + 100k + 400k + 400 = 100k^2 + 500k + 400 ).Therefore, ( F(n) = sum_{k=1}^{n} (100k^2 + 500k + 400) ).Now, I can split this summation into three separate sums:( F(n) = 100 sum_{k=1}^{n} k^2 + 500 sum_{k=1}^{n} k + 400 sum_{k=1}^{n} 1 ).I remember the formulas for these sums:1. ( sum_{k=1}^{n} k = frac{n(n + 1)}{2} ).2. ( sum_{k=1}^{n} k^2 = frac{n(n + 1)(2n + 1)}{6} ).3. ( sum_{k=1}^{n} 1 = n ).So, plugging these into the expression:First term: ( 100 times frac{n(n + 1)(2n + 1)}{6} ).Second term: ( 500 times frac{n(n + 1)}{2} ).Third term: ( 400 times n ).Let me compute each term step by step.First term:( 100 times frac{n(n + 1)(2n + 1)}{6} = frac{100}{6} n(n + 1)(2n + 1) = frac{50}{3} n(n + 1)(2n + 1) ).Second term:( 500 times frac{n(n + 1)}{2} = frac{500}{2} n(n + 1) = 250 n(n + 1) ).Third term:( 400n ).So, putting it all together:( F(n) = frac{50}{3} n(n + 1)(2n + 1) + 250 n(n + 1) + 400n ).Hmm, this looks a bit complicated. Maybe I can factor out some common terms or simplify further.Looking at the first two terms, both have ( n(n + 1) ) as a factor. Let me factor that out:( F(n) = n(n + 1) left( frac{50}{3}(2n + 1) + 250 right) + 400n ).Let me compute the expression inside the parentheses:( frac{50}{3}(2n + 1) + 250 ).First, distribute ( frac{50}{3} ):( frac{100n}{3} + frac{50}{3} + 250 ).Convert 250 to thirds to combine with ( frac{50}{3} ):250 = ( frac{750}{3} ).So, adding them together:( frac{100n}{3} + frac{50}{3} + frac{750}{3} = frac{100n + 50 + 750}{3} = frac{100n + 800}{3} ).So, now, ( F(n) = n(n + 1) times frac{100n + 800}{3} + 400n ).Let me write that as:( F(n) = frac{n(n + 1)(100n + 800)}{3} + 400n ).Factor out 100 from the numerator:( frac{n(n + 1) times 100(n + 8)}{3} + 400n = frac{100n(n + 1)(n + 8)}{3} + 400n ).Hmm, maybe I can factor out 100n or something else.Alternatively, perhaps it's better to just express it as is, but let me see if I can combine the terms.Let me write 400n as ( frac{1200n}{3} ) so that both terms have the same denominator.So, ( F(n) = frac{100n(n + 1)(n + 8)}{3} + frac{1200n}{3} ).Combine the numerators:( F(n) = frac{100n(n + 1)(n + 8) + 1200n}{3} ).Factor out 100n from the first term and 1200n as is:Wait, 100n is a common factor in both terms:Wait, 100n(n + 1)(n + 8) + 1200n = 100n [ (n + 1)(n + 8) ] + 1200n.Factor out 100n:= 100n [ (n + 1)(n + 8) + 12 ].Wait, 1200n is 12 * 100n, so:= 100n [ (n + 1)(n + 8) + 12 ].Let me compute (n + 1)(n + 8):= n^2 + 8n + n + 8 = n^2 + 9n + 8.So, adding 12:n^2 + 9n + 8 + 12 = n^2 + 9n + 20.Therefore, ( F(n) = frac{100n(n^2 + 9n + 20)}{3} ).So, simplifying:( F(n) = frac{100}{3} n(n^2 + 9n + 20) ).Alternatively, I can factor the quadratic:n^2 + 9n + 20 = (n + 4)(n + 5).So, ( F(n) = frac{100}{3} n(n + 4)(n + 5) ).That seems like a nice closed-form expression. Let me check if this makes sense.Alternatively, I can write it as:( F(n) = frac{100}{3} n(n + 4)(n + 5) ).Yes, that looks good. Let me verify with n=1.When n=1:Original sum: ( a_1 e^{b_1} = 500 e^{ln(2)} = 500 * 2 = 1000 ).Using the formula: ( frac{100}{3} *1*(1 + 4)(1 + 5) = frac{100}{3} *1*5*6 = frac{100}{3} *30 = 1000 ). Perfect, matches.Another test: n=2.Original sum:First term: 500 * 2 = 1000.Second term: a_2 = 500 + 100 = 600; e^{b_2} = 3. So, 600*3=1800.Total F(2)=1000 + 1800=2800.Using the formula: ( frac{100}{3}*2*(2 + 4)(2 + 5) = frac{100}{3}*2*6*7 = frac{100}{3}*84 = 100*28=2800 ). Perfect.Okay, so the closed-form expression is correct.Problem 2:Now, Emily starts her own foundation. The effectiveness ( b_k = k^2 ), and the grants ( a_k ) are still an arithmetic sequence with ( a_1 = 500 ) and ( d = 100 ). We need to find ( F(n) ) for the first 5 years, i.e., n=5.So, ( F(5) = sum_{k=1}^{5} a_k e^{b_k} ).First, let's compute each term individually.Compute ( a_k ) for k=1 to 5:- ( a_1 = 500 )- ( a_2 = 500 + 100 = 600 )- ( a_3 = 600 + 100 = 700 )- ( a_4 = 700 + 100 = 800 )- ( a_5 = 800 + 100 = 900 )Compute ( b_k = k^2 ), so ( e^{b_k} = e^{k^2} ).So, each term is ( a_k e^{k^2} ).Therefore, ( F(5) = 500 e^{1} + 600 e^{4} + 700 e^{9} + 800 e^{16} + 900 e^{25} ).Hmm, these exponentials are going to be huge, especially ( e^{25} ). Let me compute each term numerically.First, compute each ( e^{k^2} ):- ( e^{1} approx 2.71828 )- ( e^{4} approx 54.59815 )- ( e^{9} approx 8103.08392758 )- ( e^{16} approx 8886110.5205 )- ( e^{25} approx 7.20048993374e+10 ) (approximately 72,004,899,337.4)Now, compute each term:1. ( 500 e^{1} approx 500 * 2.71828 approx 1359.14 )2. ( 600 e^{4} approx 600 * 54.59815 approx 32,758.89 )3. ( 700 e^{9} approx 700 * 8103.08392758 approx 5,672,158.75 )4. ( 800 e^{16} approx 800 * 8,886,110.5205 approx 7,108,888,416.4 )5. ( 900 e^{25} approx 900 * 72,004,899,337.4 approx 64,804,409,403,660 )Now, sum all these up:Let me list them:1. ~1,359.142. ~32,758.893. ~5,672,158.754. ~7,108,888,416.45. ~64,804,409,403,660Adding them step by step:Start with 1,359.14 + 32,758.89 = 34,118.0334,118.03 + 5,672,158.75 = 5,706,276.785,706,276.78 + 7,108,888,416.4 ≈ 7,114,594,693.187,114,594,693.18 + 64,804,409,403,660 ≈ 64,811,524,000,000 (approximately)Wait, that seems extremely large. Let me check the calculations again.Wait, ( e^{25} ) is indeed a huge number. Let me verify the approximate value.( e^{25} ) is approximately 7.20048993374 × 10^10, which is 72,004,899,337.4.So, 900 * 72,004,899,337.4 ≈ 64,804,409,403,660.Yes, that's correct.Similarly, 800 * e^{16} ≈ 800 * 8,886,110.5205 ≈ 7,108,888,416.4.So, adding all together:1,359.14 + 32,758.89 = 34,118.0334,118.03 + 5,672,158.75 = 5,706,276.785,706,276.78 + 7,108,888,416.4 ≈ 7,114,594,693.187,114,594,693.18 + 64,804,409,403,660 ≈ 64,811,523,998,353.18So, approximately 6.481152399835318 × 10^13.But, since the problem says "determine the total fund F(n) for the first 5 years," it might be acceptable to leave it in terms of exponentials, but the question doesn't specify. It just says "determine," so maybe we can express it as the sum, but given the numbers, it's better to compute it numerically.Alternatively, perhaps the problem expects an expression rather than a numerical value because the numbers are so large. Let me see.Wait, the first part asked for a closed-form expression, and the second part just says "determine the total fund F(n) for the first 5 years." Since n=5 is small, maybe they expect the numerical value.But given that the numbers are so large, maybe they just want the expression? Let me check the original problem."2. ... determine the total fund ( F(n) ) for the first 5 years."Hmm, it doesn't specify whether to compute numerically or leave it in exponential form. Given that in the first part, they wanted a closed-form, here maybe they just want the expression, but since n=5 is small, perhaps compute it.But considering the size, it's impractical to write the exact number, so perhaps it's better to express it as the sum:( F(5) = 500e + 600e^4 + 700e^9 + 800e^{16} + 900e^{25} ).Alternatively, factor out 100:( F(5) = 100(5e + 6e^4 + 7e^9 + 8e^{16} + 9e^{25}) ).But maybe they want the numerical value. Let me see if I can compute it more accurately.Wait, let me use more precise values for the exponentials.Compute each term with more precision:1. ( e^1 approx 2.718281828459045 )2. ( e^4 approx 54.598150033144236 )3. ( e^9 approx 8103.083927575384 )4. ( e^{16} approx 8886110.520507872 )5. ( e^{25} approx 72004899337.37548 )Now, compute each term:1. ( 500 * e^1 ≈ 500 * 2.718281828459045 ≈ 1359.1409142295225 )2. ( 600 * e^4 ≈ 600 * 54.598150033144236 ≈ 32758.890019886543 )3. ( 700 * e^9 ≈ 700 * 8103.083927575384 ≈ 5672158.749302769 )4. ( 800 * e^{16} ≈ 800 * 8886110.520507872 ≈ 7108888416.406298 )5. ( 900 * e^{25} ≈ 900 * 72004899337.37548 ≈ 64804409403637.93 )Now, sum them up step by step:Start with term1 + term2:1359.1409142295225 + 32758.890019886543 ≈ 34118.030934116065Add term3:34118.030934116065 + 5672158.749302769 ≈ 5706276.780236885Add term4:5706276.780236885 + 7108888416.406298 ≈ 7114594693.186535Add term5:7114594693.186535 + 64804409403637.93 ≈ 64811523998331.116So, approximately 6.481152399833112 × 10^13.Expressed in scientific notation, that's roughly 6.4811524 × 10^13.But let me check if I can write it more precisely.Alternatively, maybe the problem expects an exact expression rather than a numerical approximation because the numbers are so large and the exact value would be unwieldy. So, perhaps the answer is best left as the sum:( F(5) = 500e + 600e^4 + 700e^9 + 800e^{16} + 900e^{25} ).But since the first part required a closed-form, maybe this part also expects a closed-form, but given the exponential terms with k^2, it's unlikely there's a simple closed-form. So, probably, the answer is the sum as above.Alternatively, factor out 100:( F(5) = 100(5e + 6e^4 + 7e^9 + 8e^{16} + 9e^{25}) ).Either way, both are acceptable. Since the problem didn't specify, but given the context, maybe the numerical value is expected, but it's so large. Alternatively, perhaps the problem expects the expression.Wait, the first part had a closed-form because the exponentials simplified to polynomials, but here, with ( b_k = k^2 ), the exponentials don't simplify, so we can't get a closed-form like in part 1. Therefore, the answer is just the sum as above.So, to conclude, for part 2, the total fund ( F(5) ) is ( 500e + 600e^4 + 700e^9 + 800e^{16} + 900e^{25} ).Alternatively, if they want the numerical value, it's approximately 6.4811524 × 10^13.But given the context, maybe they prefer the exact expression.Final Answer1. The closed-form expression for ( F(n) ) is boxed{dfrac{100}{3} n(n + 4)(n + 5)}.2. The total fund ( F(5) ) is boxed{500e + 600e^4 + 700e^9 + 800e^{16} + 900e^{25}}.</think>

🧠:<think>Alright, so I have this problem about a new treatment's effectiveness, and I need to figure out two things. First, the probability that a patient recovers in less than 10 days, and second, whether to reject the null hypothesis based on a new sample. Let me take it step by step.Starting with the first part: calculating the probability that a randomly selected patient recovers in less than 10 days. The recovery times are normally distributed with a mean of 15 days and a standard deviation of 4 days. Hmm, okay, so I remember that in a normal distribution, probabilities can be found using z-scores. The z-score formula is (X - μ) / σ, where X is the value we're interested in, μ is the mean, and σ is the standard deviation.So, plugging in the numbers: X is 10 days, μ is 15, and σ is 4. Let me compute that. 10 minus 15 is -5, and then divided by 4 is -1.25. So the z-score is -1.25. Now, I need to find the probability that Z is less than -1.25. I think this is the area to the left of -1.25 in the standard normal distribution.I remember that standard normal tables give the area to the left of a given z-score. So, I should look up -1.25 in the z-table. Let me visualize the table. The z-table has rows for the whole number and the first decimal, and columns for the second decimal. So, for -1.25, I go to the row for -1.2 and the column for 0.05. Hmm, looking that up, I think the value is around 0.1056. Wait, is that right? Let me double-check. Alternatively, I can use the symmetry of the normal distribution. The z-score of 1.25 has an area to the left of about 0.8944, so the area to the left of -1.25 would be 1 - 0.8944, which is 0.1056. Yeah, that seems correct.So, the probability that a patient recovers in less than 10 days is approximately 10.56%. That seems a bit low, but considering 10 is quite a few days below the mean of 15, it makes sense.Moving on to the second part: conducting a hypothesis test. The null hypothesis is that the mean recovery time is 15 days, and the alternative is that it's less than 15 days. We're using a 5% significance level. A new sample of 30 patients has a mean recovery time of 13.5 days. The population standard deviation is still 4 days, so we can use a z-test.Alright, so for a z-test, the formula is z = (x̄ - μ) / (σ / sqrt(n)). Let me plug in the numbers: x̄ is 13.5, μ is 15, σ is 4, and n is 30. So, 13.5 minus 15 is -1.5. The denominator is 4 divided by the square root of 30. Let me compute sqrt(30). That's approximately 5.477. So, 4 divided by 5.477 is roughly 0.7303. Therefore, the z-score is -1.5 divided by 0.7303, which is approximately -2.054.Now, since this is a one-tailed test (alternative hypothesis is less than), we need to find the p-value for a z-score of -2.054. The p-value is the probability that Z is less than -2.054. Looking at the z-table, for a z-score of -2.05, the area is about 0.0197, and for -2.06, it's about 0.0197 as well. Wait, actually, let me check more precisely. The z-table for 2.05 is 0.9798, so the area to the left is 1 - 0.9798 = 0.0202. Similarly, for 2.06, it's 0.9803, so 1 - 0.9803 = 0.0197. Since our z-score is -2.054, which is between -2.05 and -2.06, the p-value is between 0.0197 and 0.0202. To approximate, maybe around 0.02.So, the p-value is approximately 0.02, which is less than the significance level of 0.05. Therefore, we reject the null hypothesis. This suggests that the mean recovery time is significantly less than 15 days at the 5% significance level.Wait, let me make sure I didn't make a mistake in calculating the z-score. So, x̄ is 13.5, μ is 15, so difference is -1.5. Standard error is 4 / sqrt(30). Let me compute sqrt(30) more accurately. 5.477 squared is 30, right? 5.477 * 5.477 is approximately 30. So, 4 divided by 5.477 is approximately 0.7303. So, -1.5 divided by 0.7303 is indeed approximately -2.054. So, that seems correct.And the p-value for z = -2.054 is indeed around 0.02, which is less than 0.05, so we reject H0. That makes sense because the sample mean is quite a bit lower than the hypothesized mean, and with a sample size of 30, the standard error is small enough that this difference is statistically significant.So, summarizing my findings: the probability of recovering in less than 10 days is about 10.56%, and based on the new sample, we should reject the null hypothesis that the mean recovery time is 15 days in favor of the alternative that it's less than 15 days.Final Answer1. The probability is boxed{0.1056}.2. We should boxed{text{reject}} the null hypothesis.</think>

🧠:<think>Alright, so I'm trying to help this freshman student analyze the probability and impact of tornado occurrences at Wayne State College. They've given me two specific questions to tackle, both related to probability distributions. Let me break them down one by one.Starting with the first question: Calculate the probability that the student will not experience any tornadoes during their first year. They mentioned that the tornado occurrences are modeled as a Poisson distribution with an average rate of λ = 2 tornadoes per year. Hmm, okay, I remember that the Poisson distribution is used to model the number of events happening in a fixed interval of time or space, and it's characterized by the parameter λ, which is the average rate.The probability mass function for a Poisson distribution is given by P(x) = (λ^x * e^(-λ)) / x!, where x is the number of occurrences. So, in this case, we're looking for the probability that x = 0, meaning no tornadoes occur during the first year.Plugging in the numbers, we have λ = 2 and x = 0. So, P(0) = (2^0 * e^(-2)) / 0!. I know that any number raised to the power of 0 is 1, and 0! is also 1. So this simplifies to (1 * e^(-2)) / 1, which is just e^(-2). Now, e is approximately 2.71828, so e^(-2) is about 1/(e^2). Calculating e squared, 2.71828 squared is roughly 7.389. Therefore, e^(-2) is approximately 1/7.389, which is about 0.1353. So, the probability of not experiencing any tornadoes is roughly 13.53%.Wait, let me double-check that. Poisson distribution for x=0 is indeed e^(-λ), which is e^(-2). Yep, that seems right. So, the probability is e^(-2), which is approximately 0.1353 or 13.53%.Moving on to the second question: Assuming that the intensity of a tornado can be described by a continuous random variable I, following an exponential distribution with a mean intensity of 50 units on a scale, determine the expected intensity of a tornado given that one occurs during the student's first year.Okay, so here we're dealing with an exponential distribution. I recall that the exponential distribution is often used to model the time between events in a Poisson process, but in this case, it's modeling the intensity of a tornado. The probability density function (pdf) of an exponential distribution is f(x) = (1/β) * e^(-x/β) for x ≥ 0, where β is the mean of the distribution.Given that the mean intensity is 50 units, that means β = 50. So, the pdf becomes f(x) = (1/50) * e^(-x/50). The question is asking for the expected intensity of a tornado given that one occurs. Wait, but the intensity is already modeled by an exponential distribution with mean 50. So, is the expected intensity just the mean of the distribution? Because expectation is the average value we would expect.Yes, for an exponential distribution, the expected value or mean is equal to β. So, in this case, since β is 50, the expected intensity is 50 units. But hold on, is there a conditional aspect here? The question says, "given that one occurs during the student's first year." Hmm, so does that affect the expectation?Wait, the Poisson process models the number of events, and the exponential distribution models the time between events or, in this case, perhaps the intensity of each event. If the intensity is independent of the occurrence, then the expected intensity doesn't depend on whether a tornado occurs or not. It's just the inherent property of the tornado intensity distribution.So, if the intensity is exponentially distributed with mean 50, regardless of how many tornadoes occur, each tornado's intensity is independent and has an expected value of 50. Therefore, the expected intensity given that one occurs is still 50.But let me think again. Is there a scenario where the occurrence rate affects the intensity? For example, maybe more intense tornadoes are more or less likely to occur? But in the problem statement, it's given that the intensity follows an exponential distribution with a mean of 50, so I think that's independent of the Poisson process.Therefore, the expected intensity is just the mean of the exponential distribution, which is 50 units.So, to recap:1. The probability of no tornadoes is e^(-2) ≈ 0.1353.2. The expected intensity of a tornado, given that one occurs, is 50 units.I think that's it. I don't see any complications here because both distributions are independent. The Poisson models the count, and the exponential models the intensity, and they don't influence each other in this setup.Final Answer1. The probability of not experiencing any tornadoes is boxed{e^{-2}}.2. The expected intensity of a tornado is boxed{50} units.</think>

🧠:<think>Okay, so I have two mathematical problems to solve related to 3D graphics and VR systems. Let me tackle them one by one.Starting with the first problem: Surface Normal Calculation. The surface is given parametrically by the vector function r(u, v) = (u cos v, u sin v, ln u), where u > 0 and 0 ≤ v ≤ 2π. I need to find the unit normal vector N(u, v) at any point (u, v) on the surface.Hmm, I remember that for a parametric surface defined by r(u, v), the normal vector can be found by taking the cross product of the partial derivatives of r with respect to u and v. So, first, I should compute the partial derivatives.Let me write down the partial derivatives:The partial derivative of r with respect to u, which I'll call r_u, is the derivative of each component with respect to u. So:r_u = (d/du [u cos v], d/du [u sin v], d/du [ln u]) = (cos v, sin v, 1/u)Similarly, the partial derivative of r with respect to v, r_v, is:r_v = (d/dv [u cos v], d/dv [u sin v], d/dv [ln u]) = (-u sin v, u cos v, 0)Okay, so now I have r_u and r_v. The next step is to compute the cross product of r_u and r_v to get the normal vector.The cross product N = r_u × r_v can be calculated using the determinant formula:N = |i j k| |cos v sin v 1/u| |-u sin v u cos v 0|So, expanding this determinant:i * (sin v * 0 - 1/u * u cos v) - j * (cos v * 0 - 1/u * (-u sin v)) + k * (cos v * u cos v - (-u sin v) * sin v)Let me compute each component step by step.First, the i component:sin v * 0 = 01/u * u cos v = cos vSo, i * (0 - cos v) = -cos v iNext, the j component:cos v * 0 = 01/u * (-u sin v) = -sin vBut since it's subtracted, it becomes - (0 - (-sin v)) = - (sin v)Wait, hold on. The formula is:- j * (cos v * 0 - 1/u * (-u sin v)) = -j * (0 + sin v) = -j sin vWait, let me double-check:The j component is subtracted, so:- [cos v * 0 - (1/u)(-u sin v)] = - [0 + sin v] = - sin vSo, the j component is - sin v j.Now, the k component:cos v * u cos v = u cos² v(-u sin v) * sin v = -u sin² vBut since it's subtracted, it becomes:cos v * u cos v - (-u sin v) * sin v = u cos² v + u sin² vFactor out u:u (cos² v + sin² v) = u (1) = uSo, the k component is u k.Putting it all together, the normal vector N is:N = (-cos v, -sin v, u)Wait, let me confirm:i component: -cos vj component: -sin vk component: uYes, that seems right.But wait, is that the correct cross product? Let me verify the signs.The cross product formula is:If vector A = (A1, A2, A3) and vector B = (B1, B2, B3), then:A × B = (A2B3 - A3B2, A3B1 - A1B3, A1B2 - A2B1)So, applying this to r_u and r_v:r_u = (cos v, sin v, 1/u)r_v = (-u sin v, u cos v, 0)So, N_x = r_u_y * r_v_z - r_u_z * r_v_y = sin v * 0 - (1/u) * u cos v = 0 - cos v = -cos vN_y = r_u_z * r_v_x - r_u_x * r_v_z = (1/u)*(-u sin v) - cos v * 0 = -sin v - 0 = -sin vN_z = r_u_x * r_v_y - r_u_y * r_v_x = cos v * u cos v - sin v * (-u sin v) = u cos² v + u sin² v = u (cos² v + sin² v) = uYes, so N = (-cos v, -sin v, u)Okay, so that's the normal vector. But it's not necessarily a unit vector yet. I need to find the unit normal vector, so I have to divide N by its magnitude.First, let's compute the magnitude of N:|N| = sqrt( (-cos v)^2 + (-sin v)^2 + (u)^2 ) = sqrt( cos² v + sin² v + u² ) = sqrt(1 + u² )So, the unit normal vector N(u, v) is:N(u, v) = ( -cos v, -sin v, u ) / sqrt(1 + u² )Alternatively, we can write it as:N(u, v) = ( -cos v / sqrt(1 + u² ), -sin v / sqrt(1 + u² ), u / sqrt(1 + u² ) )Hmm, that seems correct. Let me just check if the cross product was correctly calculated.Wait, another way to check is to think about the orientation. Since the normal vector is pointing in a certain direction, depending on the parametrization. But in this case, since the parametrization is given, the normal vector's direction is determined by the right-hand rule of the cross product of r_u and r_v. So, as long as the cross product was computed correctly, the direction is fine.So, I think that's the unit normal vector.Moving on to the second problem: Ray-Plane Intersection.The plane is given by the equation 3x - 4y + 2z = 10. The ray originates from point P0 = (1, 2, 3) and moves in direction d = (4, -2, 1). I need to find the intersection point, if it exists.I remember that the parametric equation of a ray can be written as:P(t) = P0 + t*d, where t ≥ 0.So, substituting the given values:P(t) = (1, 2, 3) + t*(4, -2, 1) = (1 + 4t, 2 - 2t, 3 + t)Now, to find the intersection with the plane, we substitute P(t) into the plane equation:3x - 4y + 2z = 10So, substituting x = 1 + 4t, y = 2 - 2t, z = 3 + t:3*(1 + 4t) - 4*(2 - 2t) + 2*(3 + t) = 10Let me compute each term:3*(1 + 4t) = 3 + 12t-4*(2 - 2t) = -8 + 8t2*(3 + t) = 6 + 2tAdding them all together:(3 + 12t) + (-8 + 8t) + (6 + 2t) = 10Combine like terms:Constants: 3 - 8 + 6 = 1t terms: 12t + 8t + 2t = 22tSo, the equation becomes:1 + 22t = 10Subtract 1 from both sides:22t = 9So, t = 9/22 ≈ 0.4091Since t is positive, the intersection occurs at t = 9/22.Now, let's find the intersection point by plugging t back into P(t):x = 1 + 4*(9/22) = 1 + 36/22 = 1 + 18/11 = (11 + 18)/11 = 29/11 ≈ 2.636y = 2 - 2*(9/22) = 2 - 18/22 = 2 - 9/11 = (22 - 9)/11 = 13/11 ≈ 1.1818z = 3 + (9/22) = 3 + 9/22 = (66 + 9)/22 = 75/22 ≈ 3.4091So, the intersection point is (29/11, 13/11, 75/22).Wait, let me double-check the calculations:Compute x:1 + 4*(9/22) = 1 + 36/22 = 1 + 18/11 = 29/11. Correct.y:2 - 2*(9/22) = 2 - 18/22 = 2 - 9/11 = 22/11 - 9/11 = 13/11. Correct.z:3 + 9/22 = 66/22 + 9/22 = 75/22. Correct.So, the intersection point is (29/11, 13/11, 75/22). Since t is positive, the ray does intersect the plane at this point.Alternatively, if the denominator in t was zero, we would have no intersection, but in this case, the denominator was 22, which is non-zero, so intersection exists.So, summarizing:1. The unit normal vector is (-cos v, -sin v, u) divided by sqrt(1 + u²).2. The ray intersects the plane at (29/11, 13/11, 75/22).I think that's it. Let me just make sure I didn't make any arithmetic errors.For the normal vector, cross product was correct, magnitude was sqrt(1 + u²), so unit normal is as above.For the ray-plane intersection, solving for t gave a positive value, so intersection exists, and the point was correctly calculated.Yes, I think both solutions are correct.Final Answer1. The unit normal vector is (boxed{left( dfrac{-cos v}{sqrt{1 + u^2}}, dfrac{-sin v}{sqrt{1 + u^2}}, dfrac{u}{sqrt{1 + u^2}} right)}).2. The intersection point is (boxed{left( dfrac{29}{11}, dfrac{13}{11}, dfrac{75}{22} right)}).</think>