Question

The rate of population growth of pigeons in a large city is give R = 400p ^ 2 - (1/5) * p ^ 3 where p is the population of pigeons. For for what population level rate is maximized​

Answer 1

160

Explanation:

To find the population level at which the rate of population growth is maximized, we need to find the maximum value of the function R = 400p^2 - (1/5)p^3.

To do this, we can take the derivative of the function with respect to p and set it equal to 0:

R' = 800p - 3(1/5)p^2 = 0

5p = 800

p = 160

This tells us that the population level at which the rate of population growth is maximized is 160 pigeons.

To confirm that this is indeed a maximum, we can also check the second derivative of the function:

R'' = 800 - 6(1/5)p = 800 - 3p

Since the second derivative is negative when p = 160, this tells us that the function has a local maximum at p = 160. This means that the population level of 160 pigeons is indeed the population level at which the rate of population growth is maximized.