The question of whether an optimal body size exists for different kinds of animals is one that is of great interest to biologists. The reproductive power $ P $ of an individual can be modeled, following the ideas of ecologist James H. Brown (see his book *Macroecology*, University of Chicago Press, 1995), as the harmonic mean of two limiting rates. The harmonic mean of two numbers $ a $ and $ b $ is the reciprocal of the average of the inverses of the two numbers:\[\frac{1}{\frac{1}{a} + \frac{1}{b}} = \frac{ab}{a+b}\]The two rates are a per-unit-mass rate $ R_1 $ at which individuals acquire resources, and a per-unit-mass rate $ R_2 $ at which individuals convert those resources into new individuals; that is,\[P = \frac{R_1 R_2}{R_1 + R_2}\]Assuming both $ R_1 $ and $ R_2 $ are the following allometric functions of body mass measure in kilograms $ R_1 = 2M^{3/4} $ and $ R_2 = 4M^{-1/4} $,Find the body mass $ M $ that maximizes the reproductive power $ P $,\[M = \,\, \_\_\_\_\_ \,\, \text{kg}.\]

Given that, P = R1R2R1+R2 And also we have, R1= 2M3/4 and R2 = 4M-1/4 ⇒ P = 8M1/22M3/4+ 4M1/4 P = 4M1/2 . M1/4M +2 P = 4M3/42+M Now we have to find M such that the reproductive power P is maximum. That means we have to maximize P.For that we need P' and P" with respect to M, P' = (2+M)(3M-1/4)- 4M3/4(0 + 1)(2 + M )2 = 6M-1/4 + 3M3/4 - 4M3/4(2+M)2 = 6M-1/4 - M3/4(2+M)2 Now to find critical points, Consider P' = 0 So we get, 6M-1/4 = M3/4⇒ 6 = M3/4. M1/4⇒ M = 6 So the critical point of P is M = 6 Now lets find P" at M = 6, P"(M) = (2+M)2(-64M-5/4 - 34M-1/4) - (6M-1/4-M3/4)(2[2+M])(2+M)4 P"(M) = 5M2-60M - 124M5/4(2+M)3P"(6) = 180 - 360 -124. 83. (6)5/4 = -1922048.(6)5/4 < 0 Thus, we observed that P"(6) < 0 that means M = 6 is maximum of P. Thus M = 6 Kg. is the body mass that maximizes reproductive power P. So M = 6 Kg. is required answer.

Math

Calculus

The question of whether an optimal body size exists for different kinds of animals is one that is of great interest to biologists. The reproductive power P of an individual can be modeled, following the ideas of ecologist James H. Brown (see his book Macroecology, University of Chicago Press, 1995), as the harmonic mean of two limiting rates. The harmonic mean of two numbers a and b is the reciprocals of the average of the inverses of the two numbers: = ti The two rates are a per-unit-mass rate R1 at which individuals acquire resources, and a per-unit-mass rate R2 at which individuals convert those resources into new individuals; that is, RR2 P= Ri+R2 Assuming both Rị and R2 are the following allometric functions of body mass measure in kilograms R1 2M3/4 and R2 = 4M-1/4, find the body mass M that maximizes the reproductive power P, М- kg.

The question of whether an optimal body size exists for different kinds of animals is one that is of great interest to biologists. The reproductive power P of an individual can be modeled, following the ideas of ecologist James H. Brown (see his book Macroecology, University of Chicago Press, 1995), as the harmonic mean of two limiting rates. The harmonic mean of two numbers a and b is the reciprocals of the average of the inverses of the two numbers: = ti The two rates are a per-unit-mass rate R1 at which individuals acquire resources, and a per-unit-mass rate R2 at which individuals convert those resources into new individuals; that is, RR2 P= Ri+R2 Assuming both Rị and R2 are the following allometric functions of body mass measure in kilograms R1 2M3/4 and R2 = 4M-1/4, find the body mass M that maximizes the reproductive power P, М- kg.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Similar questions

Example 4: A baker for a large supermarket has received a total of 185 bags of flour from different supplier. As a result, the bags he has received are of varying weight. The histogram shows information about the weight of the bags of flour: Frequency Density 30 40 50 60 70 80 90 100 Weight (pounds) 15 bags of flour weight between 35 and 40pounds. a) How many bags of flour weigh more than 80 pounds? b) What is the median of a bag flour.
Which of the following would cause FOBT to increase A. An increase in the difference between the means b. An increase in the magnitude of the independent variables affect c. An increase in the difference between the means and an increase in the within groups variability d. An increase in the difference between the means and an increase in the magnitude of the independent variables of effect e. An increase in the within-groups variability
Please answer the whole question (2) Let x be the age of a licensed driver in years. Let y be the percentage of all fatal accidents (for a given age) due to failure to yield the right of way. For example, the first data pair states that 5% of all fatal accidents of 37-year-olds are due to failure to yield the right of way. x 37 47 57 67 77 87 y 5 8 10 15 27 44 Complete parts (a) through (e), given Σx = 372, Σy = 109, Σx2 = 24814, Σy2 = 3079, Σxy = 8043, and r ≈ 0.927. (a) Draw a scatter diagram displaying the data. Flash Player version 10 or higher is required for this question. You can get Flash Player free from Adobe's website. (b) Verify the given sums Σx, Σy, Σx2, Σy2, Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.) Σx = Σy = Σx2 = Σy2 = Σxy = r = (c) Find x, and y. Then find the equation of the least-squares line = a + bx. (Round your answers for x and y to two…
Section 3.1, question 6
Q. 3 We are given the following data for a city: Population of the city on March 30, 2020 = 183,000 Number of new active cases of TB occurring between January 1 and June 30, 2020 26 Number of active TB cases according to the city register on January 1, 2020 = 264 Find the (i) Cumulative incidence of active cases of TB for the 6-month period3; (ii) Prevalence of active TB as of June 30, 2020.
Researchers often rely on surveys to capture a subjective assessment of well-being. One study finds that holding everything else constant, people seem to be least happy when they are in their 40's. Another study suggests that money does buy happiness, but its effects diminishes as income rises above $75000 a year. The accompanying data file includes survey daa of 100 working adults' self-assessed happiness on a scale of ) to 100, along with their age and annual income. Estimate and interpret a regression model based on Age, Age 2 and income. Predict happiness with income equal = to $80000 and age equal to 30, 45, and 60 Years Predict happiness with Age equal = to 60 and income equal to 25,000, 75000, and 125000 Happiness Age Income 69 49 52000 83 47 123000 86 72 112000 73 52 166000 89 68 90000 81 37 152000 75 48 58000 56 48 50000 75 56 93000 57 51 27000 77 60 173000 83 55 173000 85 75 68000 82 69 157000 63 39 40000 70 66 107000 78 34 109000 53…
Shelia’s doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if her glucose level is above 136 milligrams per deciliter (mg/dl) one hour after ingesting a sugary drink. Shelia’s measured glucose level one hour after the sugary drink varies according to the Normal distribution with μ = 125 mg/dl and σ = 11 mg/dl. a)What are the mean and standard deviation of the distribution of sample proportion if measurements are made instead on 40 separate days? b)If measurements are made on 40 separate days and the mean result is compared with the criterion 140 mg/dl, what is the probability that Shelia is diagnosed as having gestational diabetes? c)Let’s consider what could happen if we took 40 separate measurements from Shelia. What is the blood glucose level L such that the probability…
A researcher wants to test the relationship between the number of years of formal education received (X;) and the average weekly earnings (Y;) (measured in hundred dollars). A report released by a government agency suggests that the average weekly earnings of individuals with no formal education is equal to $429. The researcher wants to test whether the average weekly earnings with no formal education is $429 or greater than that. He collects data from a sample of 100 individuals and estimates the following regression function: î, = 9.45 + 1.21X, (2.45) (4.25) where Y; is the predicted value of the weekly earnings for the " individual and the standard errors for the coefficients appear in parenthesis. The t-statistic for the test the researcher wants to conduct will be (Round your answer to two decimal places.)
Q1. A study was conducted to investigate the relationship between premature birth and mental development at 12 months. The infant's birth weight was used as an index of prematurity. At age 12 months, the child's mental development was recorded. The results are shown below (the weights are in pounds): Birth 4 6 7 8 Weight(x) Developmental 85 Quotients(y) 88 87 94 96 Summary: n = 5, £x=30, £x² = 190, Ey= 450, Ey² = 40590, Exy = 2728 . a. Find the least square prediction equation, y on x. b. Obtain a 95% confidence interval for the average value of y when x 6.5 (i.e., a C.I. for E(y|x= 6.5)). c. Obtain the coefficient of correlation (r) and interpret the result. d. Test the hypothesis that H, :p= 0 vs. H, : p0. Use a =0.05.
The table lists the average gestation period (in days) and longevity (in years) for the following mammals, as reported in The World Almanac and Book of Facts 2006. Mammal Gestation, y(days) Longevity, x(years) Mammal Gestation, y(days) Longevity, x(years) Baboon 187 20 Camel 406 12 Bear, black 219 18 Cat 63 12 Bear, grizzly 225 25 Chimpanzee 230 20 Bear, polar 240 20 Chipmunk 31 6 Beaver 105 5 Cow 284 15 Buffalo 285 15 Deer 201 8 Dog 61 12 Monkey 164 15 Donkey 365 12 Moose 240 12 Elephant 645 40 Mouse 21 3 Elk 250 15 Opossum 13 1 Fox 52 7 Pig 112 10 Giraffe 457 10 Puma 90 12 Goat 151 8 Rabbit 31 5 Gorilla 258 20 Rhinoceros 450 15 Guinea pig 48 4 Sea lion 350 12 Hippopotamus 238 41 Sheep 154 12 Horse 330 20 Squirrel 44 10 Kangaroo…
Table 9.3 Total cholesterol in 25 patients after taking new statin drug. .. Population mean is 200. Cholesterol levels 219 191 198 214 163 264 248 182 235 209 152 148 145 189 213 230 181 180 100 219 102 249 282 188 264 D. b. Describe the effects of the statin drug on your sample. Do you see any change in the level of cholesterol after taking the drug? c. What is your research question? What are the null and alternative hypotheses? Decide if you want to do a one-tailed or two-tailed test. d. What test? Perform it? What is your critical value? Sketch a normal distribution and draw the critical value or values of z. f. What is your conclusion?
Gn. Don't provide handwriting solution