Question

A population of mice is increasing exponentially. On Monday there were 120 mice. One month later there were 132 mice. Write a function to model this situation. Determine the percent increase of the population of mice each month.

Answer 1

`y =120*1.1^x`

`r = 10\%` --- Percentage increase

Explanation:

Given

Let

`x \to months`

`y \to mice`

So, we have:

`(x_1,y_1) = (0,120)` --- Monday

`(x_2,y_2) = (1,132)` --- One month later

Required

The function

The function is represented as:

`y = ab^x`

In
`(x_1,y_1) = (0,120)`, we have:

`120 =a * 1`

`120 =a`

Rewrite as:

`a = 120`

In
`(x_2,y_2) = (1,132)`, we have:

`132 = a * b^1`

`132 = a * b`

Substitute
`a = 120`

`132 = 120 * b`

Solve for b

`b = (132)/(120)`

`b = 1.1`

Recall that:

`y = ab^x`

Hence, the function is:

`y =120*1.1^x`

To get the monthly percentage increase (r), we have:

`y = a(1 + r)^x`

Compare to:
`y = ab^x`

So:

`1+r =b`

Make r the subject

`r = b-1`

`r = 1.1-1`

`r = 0.1`

Express as percentage

`r = 0.1*100\%`

`r = 10\%`