Question

Suppose a city's population grows by 5% each year. How long will it take for the population of the city to triple? Answer to the nearest hundredth of a year.

Answer 1

Since it is given that the population grows by 5% each year, it follows that the Exponential Growth Function is the appropriate function that can be used to model the problem.

The Exponential Growth Function is given by:

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

Where

• a is the initial amount.

,

• r is the percent of increase in decimal.

,

• t is the time.

,

• y is the amount after time t.

Since we want when the initial population will triple, substitute y=3a into the equation:

`3a=a(1+r)^t`

Substitute r=5%=0.05 into the equation:

`3a=a(1+0.05)^t`

Solve the resulting equation for t:

`\begin{gathered} 3a=a(1+0.05)^t \\ \Rightarrow a(1+0.05)^t=3a \\ \Rightarrow a(1.05)^t=3a \\ Divide\text{ both sides by a:} \\ \Rightarrow(1.05)^t=3 \\ \text{Take logarithm of both sides:} \\ \Rightarrow\ln (1.05)^t=\ln 3 \\ \Rightarrow t\ln (1.05)=\ln 3 \\ \Rightarrow t=(\ln 3)/(\ln (1.05))\approx22.52\text{ years} \end{gathered}`

The population will triple after about 22.52 years.