Question

The current population of a town is 10,000 and its growth in years can be represented by P(t) = 10,000(0.2)^t, where t is the time in years.

1. By what percentage does the population increase by each year?


2. Which function is correctly rewritten to reveal the monthly growth rate of the towns population?

a P(t)= 10,000(0.87449)^t

b P(t)= 10,000(0.87449)^12t

c P(t)= 10,000(0.87449)^1/t

d P(t)= 10,000(0.87449)^t+12

Please Help

Answer 1

1) 20%

2) Choice a.

Explanation:

`P(t)=10000(0.2)^t`

1)
`P(0)` is the population initially.

`P(1)` is the population after a year.

`(P(1))/(P(0))` represents the population increase factor.

So let's evaluate that fraction:

`(P(1))/(P(0))`

`(10000(0.2)^1)/(10000(0.2)^0)`

`(0.2^1)/(0.2^0)=(0.2)/(1)=0.2`

0.2=20%

2) Let's figure out the population growth in terms of months instead of years.

`P(t)=10000(0.2)^(t)`

We want t to represent months.

A full year is 12 months, in a full year we have that
`P(1)=10000(0.2)^1=10000(0.2)=2000`

So we want a new P such that
`P(12)=2000` since 12 months equals a year.

Let's look at the functions given to see which gives us this:

a)
`P(12)=10000(0.87449)^(12)=2000 \text{approximately}`

b)
`P(12)=10000(0.87449)^(12(12))=0 \text{ approximately}`

c)
`P(12)=10000(0.87449)^{(1)/(12)}=9889 \text{approximately}`

d)
`P(12)=10000(0.87449)^(12+12)=400 \text{approximately}`

So a is the function we want.

Also another way to look at this:

`P(t)=10000(.2)^t` where
`t` is in years.

`P(t)=10000(.2^(1)/(12))^t` where
`t` is in months.

And
`.2^(1)/(12)=0.874485 \text{approximately}`