Question

A city has a population of 50,000 in 2012. If the population of the city grows at an annual rate of 2%, the year in which the population will reach 100,000 is _____________ and the year it will reach 200,000 is _____________. Show work:

Answer 1

The population will reach 100,000 in 2047.

The population will reach 200,000 in 2082.

Explanation:

The compounded growth formula:

`A=P(1+r)^t`

A= Population after t years

P= Initial amount of population

r= rate of growth

t= Time in years

Given that,

P=50,000, r=2% =0.02, A=100,000

`\therefore 100,000=50,000(1+0.02)^t`

`\Rightarrow 1.02^t=(100,000)/(50,000)`

`\Rightarrow 1.02^t=2`

Taking ln both sides

`\Rightarrow ln(1.02^t)=ln|2|`

`\Rightarrow t\ ln(1.02)=ln|2|`

`\Rightarrow t=(ln|2|)/( ln(1.02))`

`\Rightarrow t\approx 35`

The population will reach 100,000 in (2012+35)=2047

P=50,000, r=2% =0.02, A=200,000

`\therefore 200,000=50,000(1+0.02)^t`

`\Rightarrow 1.02^t=(200,000)/(50,000)`

`\Rightarrow 1.02^t=4`

Taking ln both sides

`\Rightarrow ln(1.02^t)=ln|4|`

`\Rightarrow t\ ln(1.02)=ln|4|`

`\Rightarrow t=(ln|4|)/( ln(1.02))`

`\Rightarrow t\approx 70`

The year it will reach 200,000 is (2012+70)=2082