There are 20 rabbits on an island. After 6 months, the population of rabbits increased to 100. What is the monthly growth rate?

Question

Write an equation and solve.

Answer 1

Population grows exponentially with time. Therefore, you can use the equation:

P = a*b^t, where P is the population, t is the time (in months for this problem), and a and b are unknown constants.

We start with a population of 20 rabbits, which means when t=0, P = 20. We get the point (0, 20)

Plug into the equation

1) 20 = a * b⁰ = a * 1 = a

Therefore a = 20

After 6 months the population is 100, which means when t=6, P=100. We get the point (6, 100)

Plug into the equation

2) 100 = a * b⁶

We know that a = 20, so 100 = 20 * b⁶

Solving for b, we get b = ⁶√5

a represents the initial population

and b represents the monthly growth rate (it's monthly because we let t represent months)

Therefore the monthly growth rate is ⁶√5

You can think about this as every month the population multiplies by ⁶√5

Answer 2

100 = 20e^6k

5 = e^6k

ln 5 = 6k

Can you solve for k and answer?