Question

The population P(t) of worms in a compost pile after t years satisfies the logistic differential equation dP/dt =P· (2- P/2250 ) where the initial population is 18 worms.

What is the population when it's growing the fastest?

Answer 1

The population is growing the fastest when P = 2250 worms.

Step-by-step explanation:

The logistic differential equation representing the population P(t) of worms in a compost pile after t years is given by:

dP/dt = P · (2 - P/2250)

To find the population when it's growing the fastest, we need to find the value of P that maximizes the growth rate. The growth rate is given by the derivative dP/dt of the population function.

To find the critical points where the growth rate is maximized, we need to find the values of P where the derivative dP/dt is equal to zero or does not exist.

Taking the derivative of the population equation, we get:

dP/dt = 2P - (P^2/2250)

Setting this equation equal to zero and solving for P, we find:

P = 0 or P = 2250

Therefore, the population is growing the fastest when P = 2250 worms.