MHB Exponential Population Growth in Calculus

[Temit96]

The population (P) of an island y years after colonisation is given by the function
P = 250/(1+4e^-0.01y)
A. What was the initial population of the island?
B. How long did it take before the island had a population of 150?
C. After how many years was the population growing the fastest?

 

[evinda]

The population (P) of an island y years after colonisation is given by the function
P = 250/(1+4e^-0.01y)
A. What was the initial population of the island?
B. How long did it take before the island had a population of 150?
C. After how many years was the population growing the fastest?

A. You can find the initial population of the island by setting $y=0$ at the formula $P = \frac{250}{1+4e^{-0.01y}}$.

B. Set $P=150$ at the formula $P = \frac{250}{1+4e^{-0.01y}}$ and find $y$.

C. Find the derivative of $P$, then find the roots of $P'$ to find the point at which $P$ achieves its maximum.

 

[Ackbach]

For the record, our forum rules say that users asking math questions should make an effort, and show what they have done to solve the problem or answer the question. We are not a "solution factory". Instead, we find out where students are stuck, and then help them get unstuck by judicious hints. This leaves the student to do the heavy lifting, and thus they learn more.

 

[MarkFL]

For part C, you are being asked to find the maximum growth rate, not the maximum population, so you actually want to find the root(s) of $P''$. :D