Question

When does population increase the fastest for the Gompertz equation : P′ (t)=0.8ln(4000/P(t))P(t)? P=

Answer 1

In the Gompertz equation, population increases the fastest when P(t) = 4000.

Step-by-step explanation:

In the Gompertz equation, population increases fastest when the derivative of the equation, P'(t), is at its maximum. To find the maximum of P'(t), we need to find where the derivative equals zero. So, we set 0.8ln(4000/P(t))P(t) = 0 and solve for P(t).

1. Start by rearranging the equation: 0.8ln(4000/P(t))P(t) = 0
2. The natural logarithm of 1 is always 0, so we can rewrite the equation as ln(4000/P(t)) = 0
3. Raise e to the power of both sides to get 4000/P(t) = 1
4. Multiply both sides by P(t) to eliminate the denominator: 4000 = P(t)

Therefore, population increases the fastest when P(t) = 4000.