Calculus help? -WILL AWARD BEST ANSWER!?

Question

Question 1: 
Consider a predator-prey model where the prey population density is given by x, and the net growth rate of the prey population, denoted by P(x), is given by (birth rate minus rate of loss to predation)
:P(x) = rx − K(x/a + x) , where r, a, K > 0

(a) What values of x result in the smallest net growth rate? Under what condition(s) does such a minimal net growth rate exist?

(b) Suppose the number of prey is determined to be in the range 0 ≤ x ≤ N, find the absolute maximum net growth rate.

Question 2:
A cylindrical tree trunk is to be cut into a rectangular wooden beam. What is the most economical way to cut the beam so as to waste the least amount of material?

Please explain your reasoning as I wish to understand, my prof went over this weeks lesson rather poorly and I am currently studying for an exam for another class tomorrow. Thank you in advance.

cidyah · Accepted Answer

Q1.
a)
Did you mean P(x)= rx - Kx /(a+x)? If so:

P(x) = rx -K(x/(a+x))
P'(x) = r -K d/dx ( x/(a+x))

d/dx (x/(a+x)) = ((a+x)(1)-x(1)) /(a+x)^2 = (a+x-x)/(a+x)^2 = a/(a+x)^2

P'(x) = r- Ka /(a+x)^2 = 0
solve for x
r = Ka /(a+x)^2
r(a+x)^2 = Ka
(a+x)^2 = Ka/r
a+x = sqrt(Ka/r)
x = -a +sqrt(Ka/r) <-------

P''(x) = -2ka /(a+x)^3 < 0 P is maximum when x= -a +sqrt(Ka/r)

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Calculus help? -WILL AWARD BEST ANSWER!?

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