Another model for a growth function for a limited population is given by the gompertz function, which is a solution of the differential equation dpdt=cln(kp)p where c is a constant and k is the carrying capacity.(a) solve this differential equation for c=0.1, k=2000, and initial population p0=500. p(t)= .(b) compute the limiting value of the size of the population. limt→[infinity]p(t)= .(c) at what value of p does p grow fastest? p= .

A)

B) P(t→∞)=2000

C)

Step-by-step explanation:

Given differential eq is

                      --- (1)

Eq is separable

                      --- (2)

substituting in  (2)

Integrating both sides

Back substituting value of u

---(3)

at t =0

from (3)

B)

which is the carrying capacity.

C) To find the fastest growth rate we have to maximize

From given differential eq

so function to maximize is

To maximize find f'(P)=0

{x = -1/2 ,   y = -1

step-by-step explanation:

solve the following system:

{2 x + y = -2

{10 x - 3 y = -2

in the first equation, look to solve for y:

{2 x + y = -2

{10 x - 3 y = -2

subtract 2 x from both sides:

{y = -2 x - 2

{10 x - 3 y = -2

substitute y = -2 x - 2 into the second equation:

{y = -2 x - 2

{10 x - 3 (-2 x - 2) = -2

10 x - 3 (-2 x - 2) = (6 x + 6) + 10 x = 16 x + 6:

{y = -2 x - 2

{16 x + 6 = -2

in the second equation, look to solve for x:

{y = -2 x - 2

{16 x + 6 = -2

subtract 6 from both sides:

{y = -2 x - 2

{16 x = -8

divide both sides by 16:

{y = -2 x - 2

{x = -1/2

substitute x = -1/2 into the first equation:

{y = -1

{x = -1/2

collect results in alphabetical order:

answer:   {x = -1/2 ,   y = -1

b if for most of the graph

d if y intercept

step-by-step explanation: