Question

A town has a population of 20000 and grows at 4% every year. To the nearest

year, how long will it be until the population will reach 39400?

Answer 1

The population will reach 39,400 after 17 years.

Explanation:

Given that, The population grows at a rate 4%. The population of the town is 20,000 at a certain time.

Exponential function:

`y(t)=y_0(1+r)^t`

y(t) = The population after t years

`y_0` = Initial population.

r = Rate of grow

t = Time.

y(t)= 39400,
`y_0` = 20,000, r= 4%=0.04, t=?

`39400=20000(1+0.04)^t`

`\Rightarrow (39400)/(20000)=(1.04)^t`

`\Rightarrow (197)/(100)=(1.04)^t`

Taking ln both sides

`\Rightarrow ln|(197)/(100)|=ln|(1.04)^t|`

`\Rightarrow ln|(197)/(100)|=t\ ln|(1.04)|`

`\Rightarrow t= (ln|(197)/(100)|)/(ln|(1.04)|)`

`\Rightarrow t\approx 17`

The population will reach 39,400 after 17 years.