Question

A town has a population of 17000 and grows at 3% every year. To the nearest year, how long will it be until the population will reach 20400

Answer 1

Answer:

The population will reach 20,400 after 6 years.

Step-by-step explanation:

Exponential function:

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

y(t)= Population after t years

$y_0$ = initial population

r= rate of grow

t= time.

A town has a population of 17,000 and grows at a rate 3% every year.

y(t)= 20,400, $y_0$ = 17,000, r=3%=0.03, t=?

$20,400=17,000(1+0.03)^t$

$\Rightarrow \frac{20,400}{17,000}=(1.03)^t$

$\Rightarrow \frac{102}{85}=(1.03)^t$

Taking ln function both sides

$\Rightarrow ln|\frac{102}{85}|=ln|(1.03)^t|$

$\Rightarrow ln|\frac{102}{85}|=t\ ln|(1.03)|$

$\Rightarrow t=\frac{ln|\frac{102}{85}|}{\ ln|(1.03)|}$

⇒ t = 6 year.

The population will reach 20,400 after 6 years.