🧠:<think>Alright, so I've got these two math problems here about growth rates, one for a person and one for a tree. Let me try to figure them out step by step.Starting with the first problem: In 2015, a former orphan was adopted from Ukraine and moved to Florida. At the time of adoption, the orphan's height was 1.2 meters, and their growth is described by the function ( H(t) = 1.2 + ln(t + 1) ), where ( t ) is the time in years since adoption. I need to find the height in 2023.Okay, so first, let's figure out how many years have passed since 2015 until 2023. That should be 2023 minus 2015, which is 8 years. So, ( t = 8 ).Now, plugging that into the function: ( H(8) = 1.2 + ln(8 + 1) ). Simplifying inside the logarithm, that's ( ln(9) ). I remember that ( ln(9) ) is the natural logarithm of 9. Let me calculate that. I know that ( ln(9) ) is approximately 2.1972 because ( e^2 ) is about 7.389, and ( e^{2.1972} ) should be around 9. So, ( H(8) ) is approximately 1.2 + 2.1972, which is 3.3972 meters. Hmm, that seems pretty tall for someone who was 1.2 meters at 8 years old. Wait, is that realistic? Maybe, depending on their age. If they were a baby when adopted, 8 years later being almost 3.4 meters is like over 11 feet, which is way too tall. Wait, that can't be right. Maybe I made a mistake.Hold on, let's double-check. The function is ( H(t) = 1.2 + ln(t + 1) ). So, when ( t = 8 ), it's ( 1.2 + ln(9) ). Calculating ( ln(9) ) again, yeah, it's about 2.1972. So, 1.2 + 2.1972 is indeed 3.3972 meters. But that seems unrealistic for a human height. Maybe the function is supposed to be in centimeters? Wait, the problem says meters. Hmm, perhaps the function is only valid for a certain period, or maybe it's a hypothetical growth rate. Maybe it's a logarithmic growth, which is typically slower, but in this case, it's adding the logarithm to the initial height. So, maybe it's just a mathematical problem regardless of real-world plausibility.Alright, so moving on. So, the height in 2023 would be approximately 3.3972 meters, which is about 3.4 meters or 11.15 feet. I guess I'll go with that since the problem specifies the function.Now, the second problem: The adoptive family plants a tree in 2015, and the tree's height is given by ( T(t) = 0.5e^{0.3t} ), where ( t ) is the time in years since planting. I need to find the year when the tree's height surpasses the orphan's height.First, let's note that both the orphan and the tree were planted in 2015, so their time ( t ) starts at the same point. The orphan's height function is ( H(t) = 1.2 + ln(t + 1) ), and the tree's height is ( T(t) = 0.5e^{0.3t} ). I need to find the smallest ( t ) such that ( T(t) > H(t) ).So, set up the inequality: ( 0.5e^{0.3t} > 1.2 + ln(t + 1) ).This seems like a transcendental equation, meaning it can't be solved algebraically easily. I might need to use numerical methods or graphing to find the approximate value of ( t ).Let me think about how to approach this. Maybe I can compute both functions for increasing values of ( t ) until ( T(t) ) exceeds ( H(t) ).Let's start by calculating both heights at the same time points.First, at ( t = 0 ) (2015):- Orphan's height: ( H(0) = 1.2 + ln(1) = 1.2 + 0 = 1.2 ) meters.- Tree's height: ( T(0) = 0.5e^{0} = 0.5 ) meters.So, the tree is shorter.At ( t = 1 ) (2016):- Orphan: ( H(1) = 1.2 + ln(2) ≈ 1.2 + 0.6931 ≈ 1.8931 ) meters.- Tree: ( T(1) = 0.5e^{0.3} ≈ 0.5 * 1.3499 ≈ 0.67495 ) meters.Tree still shorter.At ( t = 2 ) (2017):- Orphan: ( H(2) = 1.2 + ln(3) ≈ 1.2 + 1.0986 ≈ 2.2986 ) meters.- Tree: ( T(2) = 0.5e^{0.6} ≈ 0.5 * 1.8221 ≈ 0.91105 ) meters.Still, tree is shorter.At ( t = 3 ) (2018):- Orphan: ( H(3) = 1.2 + ln(4) ≈ 1.2 + 1.3863 ≈ 2.5863 ) meters.- Tree: ( T(3) = 0.5e^{0.9} ≈ 0.5 * 2.4596 ≈ 1.2298 ) meters.Tree is still shorter.At ( t = 4 ) (2019):- Orphan: ( H(4) = 1.2 + ln(5) ≈ 1.2 + 1.6094 ≈ 2.8094 ) meters.- Tree: ( T(4) = 0.5e^{1.2} ≈ 0.5 * 3.3201 ≈ 1.66005 ) meters.Tree is still shorter.At ( t = 5 ) (2020):- Orphan: ( H(5) = 1.2 + ln(6) ≈ 1.2 + 1.7918 ≈ 2.9918 ) meters.- Tree: ( T(5) = 0.5e^{1.5} ≈ 0.5 * 4.4817 ≈ 2.24085 ) meters.Tree is still shorter.At ( t = 6 ) (2021):- Orphan: ( H(6) = 1.2 + ln(7) ≈ 1.2 + 1.9459 ≈ 3.1459 ) meters.- Tree: ( T(6) = 0.5e^{1.8} ≈ 0.5 * 6.05 ≈ 3.025 ) meters.Wait, so at ( t = 6 ), the tree is approximately 3.025 meters, and the orphan is approximately 3.1459 meters. So, the tree hasn't surpassed yet, but it's getting close.At ( t = 7 ) (2022):- Orphan: ( H(7) = 1.2 + ln(8) ≈ 1.2 + 2.0794 ≈ 3.2794 ) meters.- Tree: ( T(7) = 0.5e^{2.1} ≈ 0.5 * 8.1661 ≈ 4.08305 ) meters.Oh, wait, at ( t = 7 ), the tree is already 4.08 meters, which is taller than the orphan's 3.2794 meters. So, the tree surpasses the orphan's height between ( t = 6 ) and ( t = 7 ).To find the exact year, we need to find the exact ( t ) where ( T(t) = H(t) ). Since at ( t = 6 ), the tree is 3.025 and the orphan is 3.1459, and at ( t = 7 ), the tree is 4.083 and the orphan is 3.2794. So, the crossing point is somewhere between 6 and 7 years.Let me set up the equation: ( 0.5e^{0.3t} = 1.2 + ln(t + 1) ).This is a bit tricky to solve algebraically, so I'll use the Newton-Raphson method or trial and error to approximate the solution.Let me define ( f(t) = 0.5e^{0.3t} - 1.2 - ln(t + 1) ). We need to find ( t ) such that ( f(t) = 0 ).We know that at ( t = 6 ), ( f(6) ≈ 3.025 - 3.1459 ≈ -0.1209 ).At ( t = 7 ), ( f(7) ≈ 4.083 - 3.2794 ≈ 0.8036 ).So, the root is between 6 and 7. Let's try ( t = 6.5 ):Compute ( H(6.5) = 1.2 + ln(7.5) ≈ 1.2 + 2.015 ≈ 3.215 ) meters.Compute ( T(6.5) = 0.5e^{0.3*6.5} = 0.5e^{1.95} ≈ 0.5 * 6.8729 ≈ 3.43645 ) meters.So, ( f(6.5) ≈ 3.43645 - 3.215 ≈ 0.22145 ). So, positive.We have:- At ( t = 6 ): f ≈ -0.1209- At ( t = 6.5 ): f ≈ 0.22145So, the root is between 6 and 6.5.Let me try ( t = 6.2 ):( H(6.2) = 1.2 + ln(7.2) ≈ 1.2 + 1.974 ≈ 3.174 ) meters.( T(6.2) = 0.5e^{0.3*6.2} = 0.5e^{1.86} ≈ 0.5 * 6.444 ≈ 3.222 ) meters.So, ( f(6.2) ≈ 3.222 - 3.174 ≈ 0.048 ). Positive.At ( t = 6.1 ):( H(6.1) = 1.2 + ln(7.1) ≈ 1.2 + 1.960 ≈ 3.160 ) meters.( T(6.1) = 0.5e^{0.3*6.1} = 0.5e^{1.83} ≈ 0.5 * 6.244 ≈ 3.122 ) meters.So, ( f(6.1) ≈ 3.122 - 3.160 ≈ -0.038 ). Negative.So, between 6.1 and 6.2.At ( t = 6.15 ):( H(6.15) = 1.2 + ln(7.15) ≈ 1.2 + 1.966 ≈ 3.166 ) meters.( T(6.15) = 0.5e^{0.3*6.15} = 0.5e^{1.845} ≈ 0.5 * 6.343 ≈ 3.1715 ) meters.So, ( f(6.15) ≈ 3.1715 - 3.166 ≈ 0.0055 ). Almost zero, slightly positive.At ( t = 6.14 ):( H(6.14) = 1.2 + ln(7.14) ≈ 1.2 + 1.965 ≈ 3.165 ) meters.( T(6.14) = 0.5e^{0.3*6.14} = 0.5e^{1.842} ≈ 0.5 * 6.327 ≈ 3.1635 ) meters.So, ( f(6.14) ≈ 3.1635 - 3.165 ≈ -0.0015 ). Slightly negative.So, the root is between 6.14 and 6.15.Using linear approximation:Between ( t = 6.14 ) (f = -0.0015) and ( t = 6.15 ) (f = 0.0055). The difference in f is 0.007 over 0.01 change in t.We need to find ( t ) where f=0.The change needed from t=6.14 is (0 - (-0.0015))/0.007 = 0.0015 / 0.007 ≈ 0.2143 of the interval.So, ( t ≈ 6.14 + 0.2143*0.01 ≈ 6.14 + 0.002143 ≈ 6.1421 ) years.So, approximately 6.1421 years after 2015.Converting 0.1421 years to months: 0.1421 * 12 ≈ 1.705 months, roughly 1 month and 21 days.So, the tree surpasses the orphan's height around 6 years and 1.7 months after 2015, which would be approximately in February 2022.But since the question asks for the year, and in 2022, the tree surpasses the height, but since the exact time is around February 2022, the year would still be 2022.Wait, but in the earlier calculation, at ( t = 6.14 ), which is about 6 years and 1.7 months, so February 2022.But let me confirm with more precise calculations.Alternatively, maybe I can use a better approximation.Let me use the Newton-Raphson method.We have ( f(t) = 0.5e^{0.3t} - 1.2 - ln(t + 1) ).We need to find t where f(t)=0.We can start with an initial guess. Let's take t=6.14 where f(t)≈-0.0015.Compute f'(t) = derivative of f(t):f'(t) = 0.5*0.3e^{0.3t} - (1/(t + 1)).At t=6.14:f'(6.14) = 0.15e^{1.842} - 1/(7.14).Compute e^{1.842} ≈ 6.327, so 0.15*6.327 ≈ 0.949.1/(7.14) ≈ 0.140.So, f'(6.14) ≈ 0.949 - 0.140 ≈ 0.809.Now, Newton-Raphson update:t1 = t0 - f(t0)/f'(t0) = 6.14 - (-0.0015)/0.809 ≈ 6.14 + 0.00185 ≈ 6.14185.So, t ≈ 6.14185.Compute f(6.14185):H(t) = 1.2 + ln(7.14185) ≈ 1.2 + 1.966 ≈ 3.166.T(t) = 0.5e^{0.3*6.14185} ≈ 0.5e^{1.84255} ≈ 0.5*6.329 ≈ 3.1645.Wait, that's interesting. So, f(t) = 3.1645 - 3.166 ≈ -0.0015. Hmm, same as before.Wait, maybe my calculations are a bit off because of approximations. Alternatively, maybe my initial f(t) was slightly off.Alternatively, perhaps I should use more precise values.Alternatively, maybe it's better to use a calculator for more precise computation, but since I'm doing this manually, let's accept that the root is approximately 6.14 years.So, 6.14 years after 2015 is 2015 + 6 = 2021, plus 0.14 years.0.14 years * 12 months ≈ 1.68 months, so about 1 month and 20 days.So, approximately January 20, 2022.Therefore, the tree surpasses the orphan's height in 2022.But let me check the exact point:At t=6.14, the tree is about 3.1645 meters, and the orphan is 3.166 meters. So, the tree is just slightly shorter.At t=6.142, let's compute:T(t) = 0.5e^{0.3*6.142} = 0.5e^{1.8426} ≈ 0.5 * 6.33 ≈ 3.165 meters.H(t) = 1.2 + ln(7.142) ≈ 1.2 + 1.966 ≈ 3.166 meters.So, T(t) ≈ 3.165, H(t) ≈ 3.166. So, still, the tree is just slightly shorter.At t=6.143:T(t) = 0.5e^{0.3*6.143} = 0.5e^{1.8429} ≈ 0.5 * 6.33 ≈ 3.165 meters.H(t) = 1.2 + ln(7.143) ≈ 1.2 + 1.966 ≈ 3.166 meters.Still, the tree is just barely shorter.Wait, maybe my approximation isn't precise enough. Alternatively, perhaps the exact crossing point is around t=6.145.But regardless, it's clear that the tree surpasses the orphan's height just a bit after 6.14 years, which is still within 2022.Therefore, the year when the tree's height surpasses the orphan's height is 2022.But wait, earlier at t=6, the tree was 3.025 and the orphan was 3.1459. At t=6.14, the tree is ~3.165 and the orphan is ~3.166. So, the tree is almost equal but still slightly shorter. It might cross just a tiny bit later, but for all intents and purposes, it's 2022.Alternatively, if we consider that the tree surpasses the height in 2022, but perhaps the exact point is in early 2022, so the answer is 2022.But let me check at t=6.15:T(t)=0.5e^{1.845}=0.5*6.343≈3.1715H(t)=1.2 + ln(7.15)=1.2+1.966≈3.166So, T(t)=3.1715 > H(t)=3.166, so at t=6.15, the tree is taller.So, the crossing point is between t=6.14 and t=6.15, which is still within 2022.Therefore, the answer is 2022.But wait, let me think again. If the tree surpasses the height in 2022, but the orphan's height in 2023 is 3.3972 meters, and the tree in 2023 would be:T(8)=0.5e^{2.4}≈0.5*11.023≈5.5115 meters, which is much taller.So, the tree surpasses the orphan's height in 2022, and by 2023, it's significantly taller.So, the answer to the second question is 2022.But wait, let me make sure I didn't make a mistake in the first problem. The orphan's height in 2023 is t=8, so H(8)=1.2 + ln(9)=1.2+2.1972≈3.3972 meters. That seems correct.But just to double-check, maybe the function is supposed to be in centimeters? If so, 1.2 meters is 120 cm, and ln(9)≈2.1972 cm, so total height≈122.1972 cm, which is about 1.22 meters, which is more reasonable. But the problem says meters, so I think it's correct as is.Alternatively, maybe the function is H(t)=1.2 + ln(t+1)/100, but that's not what's written. So, I think I have to go with the given function.So, summarizing:1. The orphan's height in 2023 is approximately 3.3972 meters.2. The tree surpasses the orphan's height in 2022.But wait, in the first problem, the height is 3.3972 meters, which is about 11 feet, which is extremely tall for a human. Maybe the function is supposed to be in centimeters? Let me check the problem again.The problem says: "the orphan's height was 1.2 meters at the time of adoption and grew at a logarithmic rate described by the function ( H(t) = 1.2 + ln(t + 1) ), where ( H(t) ) is the height in meters..."So, yes, meters. So, it's correct, but it's unrealistic. Maybe it's a hypothetical scenario.Alternatively, perhaps the function is H(t)=1.2 + 0.1*ln(t+1), but that's not what's written. So, I think I have to proceed with the given function.Therefore, my final answers are:1. Approximately 3.397 meters in 2023.2. The tree surpasses the orphan's height in 2022.But let me write the exact values without rounding too much.For the first problem, H(8)=1.2 + ln(9). ln(9)=2.1972245773. So, 1.2 + 2.1972245773=3.3972245773 meters. So, approximately 3.397 meters.For the second problem, the year is 2022.So, I think that's it.</think>

🧠:<think>Okay, so I have two problems to solve here. Both are about population dynamics, one deterministic with birds and the other stochastic with beetles. Let me tackle them one by one.Starting with the first problem about the Blue-Crested Warbler. It says the population grows according to the logistic growth model. The differential equation is given as:[frac{dN}{dt} = rN left(1 - frac{N}{K}right)]where ( r = 0.3 ) per year, ( K = 500 ) birds, and the initial population ( N(0) = 50 ). I need to find the time ( t ) when the population reaches half of the carrying capacity, which would be ( N(t) = 250 ).I remember that the logistic equation has an analytical solution. Let me recall the formula. The solution to the logistic differential equation is:[N(t) = frac{K}{1 + left( frac{K - N(0)}{N(0)} right) e^{-rt}}]Yes, that seems right. So, plugging in the known values, I can solve for ( t ) when ( N(t) = 250 ).Let me write that out:[250 = frac{500}{1 + left( frac{500 - 50}{50} right) e^{-0.3 t}}]Simplify the denominator first. ( 500 - 50 = 450 ), so ( frac{450}{50} = 9 ). So the equation becomes:[250 = frac{500}{1 + 9 e^{-0.3 t}}]Let me solve for ( e^{-0.3 t} ). Multiply both sides by the denominator:[250 (1 + 9 e^{-0.3 t}) = 500]Divide both sides by 250:[1 + 9 e^{-0.3 t} = 2]Subtract 1 from both sides:[9 e^{-0.3 t} = 1]Divide both sides by 9:[e^{-0.3 t} = frac{1}{9}]Take the natural logarithm of both sides:[-0.3 t = lnleft( frac{1}{9} right)]Simplify the right side. ( ln(1/9) = -ln(9) ), so:[-0.3 t = -ln(9)]Multiply both sides by -1:[0.3 t = ln(9)]Now, solve for ( t ):[t = frac{ln(9)}{0.3}]Calculate ( ln(9) ). I know that ( ln(9) = ln(3^2) = 2 ln(3) ). Since ( ln(3) ) is approximately 1.0986, so ( 2 * 1.0986 = 2.1972 ).So,[t = frac{2.1972}{0.3} approx 7.324 text{ years}]Let me double-check my steps. Starting from the logistic equation, solving for ( t ) when ( N(t) = 250 ). Plugging into the solution formula, simplifying, taking logs. It seems correct.Now, moving on to the second problem about the beetles. The population is modeled by a stochastic differential equation (SDE):[dX_t = alpha X_t dt + sigma X_t dW_t]Given ( alpha = 0.05 ), ( sigma = 0.2 ), and ( X_0 = 100 ). I need to find the expected population ( E[X_t] ) and the variance ( Var[X_t] ) at time ( t = 5 ) years.I recall that this SDE is a geometric Brownian motion (GBM) model. The solution to this SDE is known, and the expected value and variance can be derived from it.The solution to the GBM SDE is:[X_t = X_0 expleft( left( alpha - frac{sigma^2}{2} right) t + sigma W_t right)]But for the expectation and variance, I don't need the entire distribution, just the moments.The expected value ( E[X_t] ) for GBM is:[E[X_t] = X_0 e^{alpha t}]And the variance ( Var[X_t] ) is:[Var[X_t] = X_0^2 e^{2 alpha t} left( e^{sigma^2 t} - 1 right)]Let me verify that. Yes, because the expectation of the exponential of a normal variable involves the mean and variance of the exponent. The exponent in the solution is ( (alpha - sigma^2 / 2) t + sigma W_t ). So, ( W_t ) is a Brownian motion with mean 0 and variance ( t ). Therefore, the exponent is a normal variable with mean ( (alpha - sigma^2 / 2) t ) and variance ( sigma^2 t ).Therefore, the expectation of ( X_t ) is ( X_0 e^{alpha t} ) because:[E[e^{(alpha - sigma^2 / 2) t + sigma W_t}] = e^{(alpha - sigma^2 / 2) t} E[e^{sigma W_t}] = e^{(alpha - sigma^2 / 2) t} e^{sigma^2 t / 2} = e^{alpha t}]Similarly, the variance can be calculated using the second moment. The second moment ( E[X_t^2] ) is:[E[X_t^2] = X_0^2 e^{2 (alpha - sigma^2 / 2) t} E[e^{2 sigma W_t}] = X_0^2 e^{2 alpha t - sigma^2 t} e^{2 sigma^2 t} = X_0^2 e^{2 alpha t + sigma^2 t}]Therefore, the variance is:[Var[X_t] = E[X_t^2] - (E[X_t])^2 = X_0^2 e^{2 alpha t + sigma^2 t} - (X_0 e^{alpha t})^2 = X_0^2 e^{2 alpha t} (e^{sigma^2 t} - 1)]So, plugging in the numbers.First, compute ( E[X_t] ):[E[X_t] = 100 e^{0.05 * 5} = 100 e^{0.25}]Compute ( e^{0.25} ). I remember that ( e^{0.25} ) is approximately 1.2840254.So,[E[X_t] approx 100 * 1.2840254 = 128.40254]Rounding to, say, four decimal places, it's approximately 128.4025.Next, compute the variance ( Var[X_t] ):[Var[X_t] = 100^2 e^{2 * 0.05 * 5} (e^{0.2^2 * 5} - 1) = 10000 e^{0.5} (e^{0.2} - 1)]Compute each part step by step.First, ( e^{0.5} ) is approximately 1.64872.Next, ( 0.2^2 * 5 = 0.04 * 5 = 0.2 ). So, ( e^{0.2} ) is approximately 1.221402758.Therefore, ( e^{0.2} - 1 approx 0.221402758 ).Now, compute:[10000 * 1.64872 * 0.221402758]First, multiply 1.64872 and 0.221402758:1.64872 * 0.221402758 ≈ Let me compute this.1.64872 * 0.2 = 0.3297441.64872 * 0.021402758 ≈ Approximately 1.64872 * 0.02 = 0.0329744, and 1.64872 * 0.001402758 ≈ ~0.002315So total ≈ 0.329744 + 0.0329744 + 0.002315 ≈ 0.365033So, 1.64872 * 0.221402758 ≈ 0.365033Then, multiply by 10000:0.365033 * 10000 = 3650.33So, the variance is approximately 3650.33.Let me check if that makes sense. The variance for GBM grows exponentially, so with a positive drift and volatility, it's reasonable that the variance is significantly larger than the square of the expectation.Wait, actually, let me compute the exact multiplication:1.64872 * 0.221402758Let me do it more accurately:1.64872 * 0.2 = 0.3297441.64872 * 0.02 = 0.03297441.64872 * 0.001402758 ≈ 1.64872 * 0.001 = 0.00164872, and 1.64872 * 0.000402758 ≈ ~0.000664So total ≈ 0.329744 + 0.0329744 + 0.00164872 + 0.000664 ≈ 0.365031So, 0.365031 * 10000 = 3650.31So, approximately 3650.31.Therefore, the variance is approximately 3650.31.Wait, let me compute it directly on calculator steps:First, compute ( e^{0.5} approx 1.64872 )Then, ( e^{0.2} approx 1.221402758 )So, ( e^{0.2} - 1 = 0.221402758 )Multiply 1.64872 * 0.221402758:Let me compute 1.64872 * 0.221402758:= (1 + 0.64872) * 0.221402758= 1 * 0.221402758 + 0.64872 * 0.221402758= 0.221402758 + (0.6 * 0.221402758 + 0.04872 * 0.221402758)Compute 0.6 * 0.221402758 = 0.132841655Compute 0.04872 * 0.221402758 ≈ 0.010783So total ≈ 0.221402758 + 0.132841655 + 0.010783 ≈ 0.365027So, 1.64872 * 0.221402758 ≈ 0.365027Multiply by 10000: 0.365027 * 10000 = 3650.27So, approximately 3650.27.Therefore, the variance is approximately 3650.27.So, summarizing:- Expected population ( E[X_t] approx 128.40 )- Variance ( Var[X_t] approx 3650.27 )Let me just make sure I didn't make any calculation errors.Wait, ( e^{0.25} ) is indeed approximately 1.2840254, correct.( e^{0.5} ) is approximately 1.64872, correct.( e^{0.2} ) is approximately 1.221402758, correct.Multiplying 1.64872 * 0.221402758 gives approximately 0.365027, correct.Multiply by 10000 gives 3650.27, correct.So, all steps seem correct.Therefore, the expected population after 5 years is approximately 128.40, and the variance is approximately 3650.27.I think that's all for both problems.Final Answer1. The time at which the bird population reaches half the carrying capacity is boxed{7.32} years.2. The expected beetle population after 5 years is boxed{128.40}, and the variance is boxed{3650.27}.</think>

🧠:<think>Okay, so Marie is building her family tree and she's gone back 10 generations. I need to figure out how many unique ancestors she has in the 10th generation. Hmm, let's break this down.First, each generation has an average of 2.5 children per couple. Wait, so does that mean each couple has 2.5 kids on average? So, if we think about it, each parent would have 2.5 children. But when it comes to ancestors, each person has two parents, right? So, for each generation going back, the number of ancestors doubles. But here, the average number of children is 2.5, which is a bit more than 2. So, does that affect the number of ancestors?Wait, no, maybe I'm confusing something. Let me think again. The average number of children per couple is 2.5. So, each couple has 2.5 children on average. But for ancestors, each person has two parents, regardless of how many children the couple has. So, actually, the number of ancestors per generation should still double each time, right? Because each person has two parents, four grandparents, eight great-grandparents, and so on.But hold on, the problem says "using an exponential growth model" and mentions that each generation follows the pattern of the average number of children per couple. Hmm, maybe I need to model the number of ancestors based on the average number of children. Let me clarify.If each couple has 2.5 children on average, then each child has two parents. So, for each generation going back, the number of ancestors should multiply by 2.5? Wait, that doesn't sound right. Because each person has two parents, so the number of ancestors should double each generation, regardless of the number of children per couple. The number of children per couple affects the number of descendants, not ancestors.Wait, maybe I'm overcomplicating. Let's think about it step by step.In the first generation (her parents), there are 2 people.In the second generation (her grandparents), each parent has two parents, so 4 people.Third generation (great-grandparents): 8 people.And so on, doubling each time.So, for the nth generation, the number of ancestors is 2^n.So, for the 10th generation, it would be 2^10, which is 1024.But wait, the problem mentions an average of 2.5 children per couple. Does that affect the number of ancestors? Because if each couple has 2.5 children, does that mean that each person has 2.5 parents? That doesn't make sense because you can't have half a parent.Alternatively, maybe it's about the number of descendants. Each couple has 2.5 children on average, so each generation has 2.5 times the number of people as the previous generation. But that's about descendants, not ancestors.Wait, the question is about ancestors, so maybe the number of ancestors is still doubling each generation, regardless of the number of children per couple. Because each person has two parents, four grandparents, etc.So, perhaps the 2.5 is a red herring here, or maybe it's used in a different way.Wait, let me read the problem again: "Calculate the total number of unique ancestors Marie would have in the 10th generation... using an exponential growth model. Assume each generation follows the pattern of the average number of children per couple."Hmm, so maybe it's not just doubling each time, but using the average number of children per couple as the growth factor.Wait, so if each couple has 2.5 children, then each generation is 2.5 times the previous generation. But that would be for descendants, not ancestors. For ancestors, it's the inverse.Wait, maybe the number of ancestors is multiplied by 2 each generation, but the number of descendants is multiplied by 2.5 each generation.So, if we're talking about ancestors, it's still 2^n, but if we were talking about descendants, it's 2.5^n.But the problem says "using an exponential growth model" and "each generation follows the pattern of the average number of children per couple." So maybe it's referring to the number of ancestors growing exponentially with the rate of 2.5 per generation?Wait, that doesn't make sense because each person has two parents, so the number of ancestors should be 2^n, not 2.5^n.I'm confused now. Let me think differently.If each couple has 2.5 children, then the number of children per couple is 2.5. So, for each person, how many parents do they have? Two, right? So, the number of ancestors per generation is still doubling each time.Therefore, the number of ancestors in the 10th generation should be 2^10, which is 1024.But why does the problem mention the average number of children per couple? Maybe it's a distractor, or maybe it's used in a different way.Wait, perhaps the question is about the total number of unique ancestors across all 10 generations, but the way it's phrased is "in the 10th generation." So, it's specifically asking for the number of ancestors in the 10th generation, not the total across all generations.So, in that case, it's just 2^10, which is 1024.But let me make sure. If each generation has 2.5 children per couple, does that mean that each person has 2.5 parents? That can't be. Each person has two parents, regardless of how many children the couple has.So, the number of ancestors in each generation is still doubling. So, 10th generation would be 2^10 = 1024.Okay, so maybe the 2.5 is just extra information, or perhaps it's used in a different way that I'm not seeing.Alternatively, maybe the problem is considering that each couple has 2.5 children, so the number of couples in each generation is increasing by 2.5 times? Wait, no, because each couple has 2.5 children, but each child comes from a couple, so the number of couples in the next generation would be equal to the number of children divided by 2, since each couple has two parents.Wait, this is getting too convoluted. Let me try to model it.Let's say in generation 1 (Marie's generation), there is 1 person.Generation 2 (her parents): 2 people.Generation 3 (her grandparents): each parent has 2.5 children, but wait, that's about descendants, not ancestors.Wait, maybe the number of ancestors is calculated based on the number of children per couple. So, if each couple has 2.5 children, then each child has two parents, but the number of ancestors per generation is multiplied by 2.5? That doesn't make sense because you can't have half an ancestor.Wait, perhaps it's the other way around. If each couple has 2.5 children, then each child has two parents, but the number of unique ancestors is increasing by a factor related to 2.5.Wait, I'm getting stuck here. Maybe I should look for a formula or think about it differently.In general, the number of ancestors in the nth generation is 2^n, assuming no intermarriage and each person has two unique parents.But if the average number of children per couple is 2.5, does that affect the number of ancestors? Or is it just about the number of descendants?Wait, maybe the problem is trying to say that each generation has 2.5 times the number of people as the previous generation, but that would be for descendants, not ancestors.Wait, let's think about it as a family tree. Each person has two parents, so each generation going back doubles the number of ancestors. So, generation 1: 1, generation 2: 2, generation 3: 4, generation 4: 8, and so on. So, generation n: 2^(n-1).Wait, but the problem says "the 10th generation," so that would be 2^10 = 1024.But the problem mentions an average of 2.5 children per couple. Maybe it's trying to say that each couple has 2.5 children, so the number of ancestors is actually 2.5^n? But that would be for descendants.Wait, no, because each person has two parents, regardless of how many children the couple has. So, the number of ancestors should still be 2^n.Wait, maybe the problem is considering that each couple has 2.5 children, so each child has two parents, but the number of unique ancestors is 2.5 times the previous generation? That doesn't make sense because you can't have half an ancestor.Wait, perhaps the problem is using the average number of children per couple to model the growth of the family tree, but since we're going back in generations, it's the inverse. So, if each couple has 2.5 children, then each child has two parents, but the number of ancestors would be multiplied by 2 each time, regardless of the number of children.So, maybe the 2.5 is just extra information, or perhaps it's a red herring.Alternatively, maybe the problem is considering that each couple has 2.5 children, so the number of couples in each generation is increasing by 2.5 times, but that would be for descendants, not ancestors.Wait, I'm going in circles here. Let me try to think of it as a branching process.In a branching process, each individual has a certain number of offspring. Here, each couple has 2.5 children on average. So, the offspring per couple is 2.5.But for ancestors, it's the inverse. Each individual has two parents. So, the number of ancestors doubles each generation.Therefore, the number of ancestors in the 10th generation is 2^10 = 1024.So, maybe the answer is 1024.But let me check if the average number of children per couple affects this. If each couple has 2.5 children, does that mean that each person has 2.5 parents? No, that's not possible. Each person has two parents, regardless of how many children the couple has.Therefore, the number of ancestors in each generation is still 2^n, where n is the generation number.So, for the 10th generation, it's 2^10 = 1024.Okay, I think that's the answer for part 1.Now, moving on to part 2.Given that the probability of any two people in France being related within 10 generations is 1 in 100,000, determine the probability that one of Marie's 10th generation ancestors is not related to a randomly chosen person from the current French population of 67 million people.Hmm, so the probability that two people are related within 10 generations is 1/100,000. So, the probability that they are not related is 1 - 1/100,000 = 99,999/100,000.But Marie has 1024 ancestors in the 10th generation. So, we need to find the probability that none of these 1024 ancestors are related to a randomly chosen person.Wait, but the probability is given for any two people. So, if we have 1024 ancestors, the probability that at least one of them is related to the random person is 1 minus the probability that none of them are related.But wait, the events are not independent, because if one ancestor is related, it affects the probability of another ancestor being related. But given that the probability is very low (1/100,000), maybe we can approximate it using the Poisson approximation or something.Alternatively, since the probability is so low, the probability that none of the 1024 ancestors are related is approximately (1 - 1/100,000)^1024.But let's calculate that.First, the probability that a single ancestor is not related is 1 - 1/100,000 = 0.99999.So, the probability that none of the 1024 ancestors are related is (0.99999)^1024.We can approximate this using the formula (1 - x)^n ≈ e^(-nx) when x is small.So, (0.99999)^1024 ≈ e^(-1024 * 1/100,000) = e^(-1024/100,000) = e^(-0.01024).Now, e^(-0.01024) is approximately 1 - 0.01024 + (0.01024)^2/2 - ... ≈ 0.9898.But let's calculate it more accurately.We know that ln(0.99999) ≈ -0.00001, so ln((0.99999)^1024) = 1024 * ln(0.99999) ≈ 1024 * (-0.00001) = -0.01024.So, (0.99999)^1024 ≈ e^(-0.01024) ≈ 0.9898.Therefore, the probability that none of Marie's 10th generation ancestors are related to the random person is approximately 0.9898.But wait, the question is asking for the probability that one of Marie's ancestors is NOT related. So, that's the same as the probability that none of them are related, which we just calculated as approximately 0.9898.But let me double-check.The probability that a single ancestor is not related is 0.99999.The probability that all 1024 ancestors are not related is (0.99999)^1024 ≈ 0.9898.So, the probability that at least one ancestor is related is 1 - 0.9898 = 0.0102, or about 1.02%.But the question is asking for the probability that one of Marie's ancestors is NOT related. Wait, no, it's the probability that one of her ancestors is not related. Wait, no, the wording is: "the probability that one of Marie's 10th generation ancestors is not related to a randomly chosen person."Wait, actually, no. The wording is: "the probability that one of Marie's 10th generation ancestors is not related to a randomly chosen person."Wait, that's a bit ambiguous. Does it mean the probability that at least one ancestor is not related, or the probability that a randomly chosen ancestor is not related?Wait, let's read it again: "the probability that one of Marie's 10th generation ancestors is not related to a randomly chosen person."Hmm, it's a bit unclear. It could be interpreted as the probability that at least one ancestor is not related, but that's almost certain because the probability of being related is 1/100,000, so the chance that all 1024 are related is extremely low.Alternatively, it could be interpreted as the probability that a randomly selected ancestor is not related, which would be 1 - 1/100,000 = 0.99999.But I think the intended interpretation is the probability that at least one of the ancestors is not related, but that's almost 1, which doesn't make sense given the answer format.Wait, maybe it's the probability that a randomly chosen person is not related to any of Marie's ancestors. So, the probability that the person is not related to Marie's family tree.In that case, it's the same as the probability that none of Marie's 1024 ancestors are related to the person, which we calculated as approximately 0.9898.But let me think again.The problem says: "the probability that one of Marie's 10th generation ancestors is not related to a randomly chosen person."Wait, that could be interpreted as the probability that, among Marie's ancestors, at least one is not related to the random person. But since the probability of being related is 1/100,000, the probability that a specific ancestor is not related is 99,999/100,000. So, the probability that at least one ancestor is not related is 1 minus the probability that all ancestors are related.But the probability that all 1024 ancestors are related to the random person is (1/100,000)^1024, which is practically zero. So, the probability that at least one ancestor is not related is approximately 1.But that seems too straightforward, and the answer would be almost 1, which is 100%.But that doesn't seem right because the question is probably asking for the probability that a randomly chosen person is not related to any of Marie's ancestors, which is the same as the probability that none of Marie's ancestors are related to the person, which we calculated as approximately 0.9898.Wait, maybe I'm overcomplicating. Let's think of it as the probability that the random person is not related to any of Marie's 1024 ancestors. So, it's the same as the probability that none of the 1024 ancestors are related to the person.So, that would be (1 - 1/100,000)^1024 ≈ e^(-1024/100,000) ≈ e^(-0.01024) ≈ 0.9898.Therefore, the probability is approximately 0.9898, or 98.98%.But let me express it more precisely.We can calculate (1 - 1/100,000)^1024 exactly using logarithms.Take natural log: ln((1 - 1/100,000)^1024) = 1024 * ln(1 - 0.00001) ≈ 1024 * (-0.00001000005) ≈ -0.0102400512.So, exponentiate: e^(-0.0102400512) ≈ 0.9898058.So, approximately 0.9898, or 98.98%.Therefore, the probability is approximately 0.9898, or 98.98%.But let me check if the question is asking for the probability that one of Marie's ancestors is not related, which could be interpreted as the probability that at least one ancestor is not related, which is 1 - probability that all ancestors are related.But the probability that all ancestors are related is (1/100,000)^1024, which is practically zero. So, the probability that at least one ancestor is not related is approximately 1.But that contradicts the earlier interpretation. So, I think the correct interpretation is the probability that the random person is not related to any of Marie's ancestors, which is approximately 0.9898.Therefore, the answer is approximately 0.9898, or 98.98%.But let me express it as a fraction or a more precise decimal.Since 1 - 1/100,000 = 0.99999, and we have 1024 ancestors, the probability is (0.99999)^1024.We can calculate this more accurately.Using the formula (1 - x)^n ≈ e^(-nx) for small x.Here, x = 1/100,000 = 0.00001, n = 1024.So, e^(-1024 * 0.00001) = e^(-0.01024) ≈ 0.9898058.So, approximately 0.9898, or 98.98%.Therefore, the probability is approximately 0.9898, or 98.98%.But let me check if I can express this as a fraction.Since 1/100,000 is 0.00001, and 1024 * 0.00001 = 0.01024.So, e^(-0.01024) ≈ 1 - 0.01024 + (0.01024)^2/2 - (0.01024)^3/6 + ...Calculating up to the second term: 1 - 0.01024 = 0.98976.Adding the third term: (0.01024)^2 / 2 = 0.0001048576 / 2 = 0.0000524288.So, 0.98976 + 0.0000524288 ≈ 0.9898124288.Subtracting the fourth term: (0.01024)^3 / 6 ≈ 0.000001073741824 / 6 ≈ 0.00000017895697.So, 0.9898124288 - 0.00000017895697 ≈ 0.9898122498.So, approximately 0.989812.Therefore, the probability is approximately 0.9898, or 98.98%.So, rounding to four decimal places, it's 0.9898.But let me check if I can express it as a fraction.Since 0.9898 is approximately 9898/10000, but that's not simplified.Alternatively, 0.9898 is approximately 9898/10000, which simplifies to 4949/5000, but that's not exact.Alternatively, we can leave it as approximately 0.9898.Therefore, the probability is approximately 0.9898, or 98.98%.So, summarizing:1. The number of unique ancestors in the 10th generation is 2^10 = 1024.2. The probability that a randomly chosen person is not related to any of Marie's 10th generation ancestors is approximately 0.9898, or 98.98%.But let me make sure about part 2 again.The probability that two people are related is 1/100,000. So, the probability that they are not related is 99,999/100,000.Marie has 1024 ancestors. The probability that none of them are related to the random person is (99,999/100,000)^1024.Which is approximately e^(-1024/100,000) ≈ e^(-0.01024) ≈ 0.9898.So, yes, that's correct.Therefore, the answers are:1. 10242. Approximately 0.9898, or 98.98%</think>

🧠:<think>Alright, so I have this problem about an ice core sample from Antarctica. It's cylindrical with a radius of 0.05 meters and a length of 20 meters. The geochemist is looking at the concentration of this isotope called Isotope X, and there's a function given for its concentration at depth z: C(z) = 5 + 2cos(πz/4) + (1/2)sin(πz/2). The first part is to calculate the total mass of Isotope X in the ice core. They also mention the density of ice is 900 kg/m³, and Isotope X contributes negligibly to the mass, so I can ignore its contribution when calculating the total mass of the ice. Okay, so to find the total mass of Isotope X, I need to find the volume of the ice core and then integrate the concentration over that volume. Since the ice core is cylindrical, the volume can be found using the formula for the volume of a cylinder: V = πr²L. Given r = 0.05 m and L = 20 m, so plugging in those values: V = π*(0.05)²*20. Let me compute that. First, (0.05)² is 0.0025. Multiply that by 20, which gives 0.05. Then multiply by π, so V ≈ 0.05π m³. That's approximately 0.15708 m³. But wait, actually, I think I made a mistake here. Let me recalculate. Radius r = 0.05 m, so area A = πr² = π*(0.05)^2 = π*0.0025 ≈ 0.007854 m². Then, volume V = A*L = 0.007854*20 ≈ 0.15708 m³. Yeah, that's correct. So the total volume is approximately 0.15708 cubic meters.But wait, actually, for the mass, I need to consider the density. The density is 900 kg/m³, so the total mass of the ice is density multiplied by volume: 900 kg/m³ * 0.15708 m³ ≈ 141.372 kg. But since Isotope X contributes negligibly, the mass of Isotope X will be a small fraction of this. But to find the total mass of Isotope X, I need to integrate the concentration over the volume. Since concentration is given as a function of depth z, and the ice core is uniform in radius, the concentration only varies with z. So, I can set up an integral along the depth z from 0 to 20 meters. The mass of Isotope X, m, would be the integral from z=0 to z=20 of (concentration C(z) * density ρ * cross-sectional area A) dz. Wait, no. Actually, concentration is in ppm, which is parts per million by mass. So, if I have the concentration in ppm, then the mass of Isotope X is (C(z) / 1,000,000) * total mass of ice at depth z. But since the ice core is uniform in radius, the mass per unit depth is constant. So, the mass per unit depth (dm/dz) is density * area * dz. So, dm = ρ * A * dz. Therefore, the mass of Isotope X at depth z is (C(z)/1,000,000) * dm. So, the total mass M is the integral from 0 to 20 of (C(z)/1,000,000) * ρ * A dz. Alternatively, since C(z) is in ppm, which is mass fraction, so M = ∫ (C(z) * ρ * A) dz / 1,000,000. Wait, let me think carefully. If C(z) is ppm by mass, then the mass of Isotope X is (C(z) / 1,000,000) * (mass of ice at depth z). The mass of ice at depth z is ρ * A * dz, so M = ∫ (C(z)/1,000,000) * ρ * A dz from 0 to 20. Alternatively, since A is constant, I can factor that out: M = (ρ * A / 1,000,000) * ∫ C(z) dz from 0 to 20. Yes, that seems correct. So, let me compute that. First, compute ρ * A / 1,000,000. ρ = 900 kg/m³, A = π*(0.05)^2 ≈ 0.007854 m². So, 900 * 0.007854 ≈ 7.0686 kg/m. Then, divide by 1,000,000: 7.0686 / 1,000,000 ≈ 7.0686e-6 kg/m. So, M = 7.0686e-6 * ∫ C(z) dz from 0 to 20. So, I need to compute the integral of C(z) from 0 to 20. Let's write out C(z):C(z) = 5 + 2cos(πz/4) + (1/2)sin(πz/2)So, ∫ C(z) dz = ∫ [5 + 2cos(πz/4) + (1/2)sin(πz/2)] dz from 0 to 20.Let me compute each integral separately.First, ∫5 dz from 0 to 20 is 5z evaluated from 0 to 20, which is 5*20 - 5*0 = 100.Second, ∫2cos(πz/4) dz. The integral of cos(ax) is (1/a)sin(ax). So, 2*(4/π) sin(πz/4) evaluated from 0 to 20. So, 8/π [sin(5π) - sin(0)]. Sin(5π) is 0, sin(0) is 0, so this integral is 0.Third, ∫(1/2)sin(πz/2) dz. The integral of sin(ax) is -(1/a)cos(ax). So, (1/2)*(-2/π) cos(πz/2) evaluated from 0 to 20. Simplify: (-1/π)[cos(10π) - cos(0)]. Cos(10π) is 1, cos(0) is 1. So, (-1/π)(1 - 1) = 0.So, the total integral ∫ C(z) dz from 0 to 20 is 100 + 0 + 0 = 100.Therefore, M = 7.0686e-6 * 100 ≈ 7.0686e-4 kg. Convert that to grams: 7.0686e-4 kg = 0.70686 grams.Wait, but let me double-check the integral calculations because sometimes the periodic functions can have non-zero integrals over certain intervals.Wait, for the second integral: ∫2cos(πz/4) dz from 0 to 20. The period of cos(πz/4) is 8 meters because period T = 2π/(π/4) = 8. So, over 20 meters, which is 2.5 periods. So, integrating over an integer number of periods would give zero, but 2.5 is not an integer. Wait, but in our calculation, we found that sin(5π) is zero, so the integral is zero. Hmm, that's correct because sin(nπ) is zero for integer n. So, 5π is 5 times π, which is an integer multiple, so sin(5π)=0. So, that integral is indeed zero.Similarly, for the third integral: ∫(1/2)sin(πz/2) dz from 0 to 20. The period of sin(πz/2) is 4 meters. So, over 20 meters, that's 5 periods. So, integrating over an integer number of periods would give zero. And indeed, we found that cos(10π) - cos(0) = 1 - 1 = 0, so the integral is zero.Therefore, the total integral is indeed 100. So, M ≈ 7.0686e-4 kg, which is 0.70686 grams. But let me check the units again. The concentration is in ppm, which is mass per mass. So, when I do (C(z)/1e6) * mass of ice, that gives the mass of Isotope X. But wait, in the integral, I set M = (ρ * A / 1e6) * ∫ C(z) dz. Let me verify the units:ρ is kg/m³, A is m², so ρ*A is kg/m. Then, dz is in meters, so ρ*A*dz is kg. So, (ρ*A*dz)/1e6 is kg, and multiplied by C(z) which is ppm (dimensionless), so the units are correct.Alternatively, another way: The mass of Isotope X is ∫ (C(z) / 1e6) * (ρ * A dz). So, yes, that's correct.So, the total mass is approximately 0.70686 grams. But let me compute it more precisely. First, compute ρ * A: 900 kg/m³ * π*(0.05)^2 m². Compute 0.05 squared: 0.0025. Multiply by π: ≈0.00785398 m². Then, 900 * 0.00785398 ≈ 7.068582 kg/m. Then, divide by 1e6: 7.068582e-6 kg/m. Multiply by the integral of C(z) from 0 to 20, which is 100: 7.068582e-6 * 100 = 7.068582e-4 kg. Convert to grams: 7.068582e-4 kg = 0.7068582 grams. So, approximately 0.7069 grams. But perhaps we can write it more accurately. Let me compute 7.068582e-4 kg: that's 0.0007068582 kg, which is 0.7068582 grams. So, rounding to a reasonable number of decimal places, maybe 0.7069 grams or 0.707 grams.Alternatively, if we need to express it in scientific notation, it's 7.068582e-4 kg, but grams is more intuitive here.So, the total mass of Isotope X is approximately 0.707 grams.Wait, but let me think again. Is there another way to approach this? Maybe by considering the average concentration and then multiplying by the total mass of the ice.The average concentration would be (1/20) ∫ C(z) dz from 0 to 20, which is 100/20 = 5 ppm. So, average concentration is 5 ppm.Total mass of ice is ρ * V = 900 kg/m³ * 0.15708 m³ ≈ 141.372 kg. Then, mass of Isotope X is 5 ppm of that: 141.372 kg * (5 / 1e6) = 141.372 * 5e-6 ≈ 0.00070686 kg ≈ 0.70686 grams. So, same result. So, that's a good check. So, the total mass is approximately 0.707 grams.So, that's part 1 done.Now, part 2: The geochemist models the temperature T(z) as T(z) = T0 + kC(z), where T0 = -30°C and k = 0.1°C/ppm. We need to find the average temperature across the entire depth of the ice core.So, average temperature would be (1/L) ∫ T(z) dz from 0 to L, where L=20 m.So, average T = (1/20) ∫ [T0 + kC(z)] dz from 0 to 20.We can split this into two integrals: (1/20)[ ∫ T0 dz + k ∫ C(z) dz ] from 0 to 20.We already computed ∫ C(z) dz from 0 to 20 as 100. So, ∫ T0 dz from 0 to 20 is T0*(20 - 0) = -30*20 = -600.So, average T = (1/20)[ -600 + k*100 ].Given k = 0.1°C/ppm, so k*100 = 0.1*100 = 10.Therefore, average T = (1/20)(-600 + 10) = (1/20)(-590) = -590/20 = -29.5°C.So, the average temperature is -29.5°C.Wait, let me verify that. T(z) = -30 + 0.1*C(z). Average T = (1/20) ∫ (-30 + 0.1*C(z)) dz from 0 to 20.= (1/20)[ ∫-30 dz + 0.1 ∫C(z) dz ]= (1/20)[ -30*20 + 0.1*100 ]= (1/20)[ -600 + 10 ]= (1/20)(-590) = -29.5°C.Yes, that's correct.Alternatively, since we know the average concentration is 5 ppm, as computed earlier, then average T = T0 + k*(average C) = -30 + 0.1*5 = -30 + 0.5 = -29.5°C. Same result. So, that's a good check.So, the average temperature is -29.5°C.Therefore, the answers are:1. Total mass of Isotope X: approximately 0.707 grams.2. Average temperature: -29.5°C.But let me write the exact values without rounding.For part 1, the integral of C(z) was exactly 100, so M = (900 * π * 0.05² / 1e6) * 100.Compute 900 * π * 0.0025 / 1e6 * 100.Wait, 900 * π * 0.0025 = 900 * 0.00785398 ≈ 7.068582.Then, 7.068582 / 1e6 * 100 = 7.068582e-4 kg, which is 0.7068582 grams.So, exact value is 0.7068582 grams. If we want to express it more precisely, maybe keep more decimal places, but 0.707 grams is sufficient.For part 2, the average temperature is exactly -29.5°C.So, summarizing:1. Total mass of Isotope X: 0.707 grams.2. Average temperature: -29.5°C.Final Answer1. The total mass of Isotope X is boxed{0.707 text{ grams}}.2. The average temperature across the ice core is boxed{-29.5^circ text{C}}.</think>