Question

An animal population is modeled by the function A(t) that satisfies the differential equation dA dt equals the product of A divided by 1125 and the quantity 450 minus A . What is the animal population when the population is increasing most rapidly?

45 animals 225 animals 40 animals 180 animals

Answer 1

225 animals

Explanation:

From the given information:

`(dA)/(dt) = ( (A)/(1125)) (450 - A)`

`(dA)/(dt) = (450A - A^2)/(1125)`

For a population increasing most rapidly; we have:

`((dA)/(dt) \implies max = A')`

Thus; for
`(d)/(dA) ( (dA)/(dt)) \implies (450-2A)/(1125)`

`(dA')/(dA) \implies 0 \\ \\ \\ A = 225`