Question

An endangered species of marsupials has a population of 178. However, with a successful breeding program it is expected to increase by 32% each year.

a Find the expected population size after:
i 10 years
it 25 years.
b How long will it take for the population to reach 10 000?

Answer 1

P = 178 (initial population size)

r = 32% = 0.32 (growth rate as a decimal)

n = 10 (number of years)

A = P(1 + r)^n

A = 178(1 + 0.32)^10

A = 178(1.32)^10

A ≈ 178(16.05976966)

A ≈ 2858.639

The expected population size after 10 years is approximately 2859.

Let's calculate the expected population size after 25 years with a 32% annual growth rate:

P = 178 (initial population size)

r = 32% = 0.32 (growth rate as a decimal)

n = 25 (number of years)

A = P(1 + r)^n

A = 178(1 + 0.32)^25

A = 178(1.32)^25

A ≈ 178(1033.589955)

A ≈ 183,979.0121

The expected population size after 25years is approximately 183,979.

To find out how long it will take for the population to reach 10,000, we need to solve for n:

P = 178

r = 32% = 0.32

A = 10,000

10,000 = 178(1 + 0.32)^n

Dividing both sides by 178:

5.592841163 = (1.32)^n

Taking the logarithm of both sides (base 10 or natural logarithm can be used):

log(5.592841163 ) = n * log(1.32)

Using logarithm properties, we can isolate n:

n = log(5.592841163 ) / log(1.32)

n ≈ 6.200614661

So, it will take approximately 6.20 years for the population to reach 10,000.