Question

The human population of a remote island is given by the formula p=kr^t the Where t is the time in years from when records were first kept and k and r are values to be determined. Find these values given the following information. the population is initially 1000. the population is 1200 after one year. What is the population after 10 years?

Answer 1

Part 1)
`k=1,000,r=1.2,p=1,000(1.2^t)`

Part 2) The population after 10 years is about 6,192 humans

Explanation:

Part 1) Find the values of k and r

Let

p ----> the human population of a remote island

t ----> the time in years

we have

`p=k(r^t)`

This is a exponential function of the form
`y=a(b^x)`

where

a is the initial value or y-intercept

b is the base of the exponential function

In this problem

The initial population is 1,000

so

`k=1,000`

`p=1,000(r^t)`

Remember that

the population is 1200 after one year

we have the ordered pair (1,1,200)

substitute in the equation and solve for r

`1,200=1,000(r^1)\\1,000r=1,200\\r=1.2`

therefore

`p=1,000(1.2^t)`

Part 2) What is the population after 10 years?

For x=10 years

substitute the value of x in the exponential function

`p=1,000(1.2^(10))\\p=6,192`

therefore

The population after 10 years is about 6,192 humans