Question

There are 6500 people in a certain area. The area is growing at a rate of 3.5%. Use the formula A=Pertto calculate the time it would take for the area to reach 11500 people. Calculate the solution for t to the nearest tenth using logarithms.

Answer 1

ANSWER

t = 16.6

EXPLANATION

In the formula:

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

A is the final amount, in this case is 11500 people. P is the initial amount, in this case is 6500 people. r is the growth rate, which in this case is 0.035 - we have to express it as a decimal, not a percent. And finally t is the time, which is the variable we want to find.

Let's clear t first and replace all these values at the end. To clear t we have to leave only the factor which exponent is t on one side of the equation. To do this we have to divide both sides by P:

$\begin{gathered} \frac{A}{P}=\frac{P}{P}(1+r)^t \\ \frac{A}{P}=(1+r)^t \end{gathered}$

Now we have to use the following rule for the exponents and logarithms:

$\begin{gathered} a^x=b \\ \text{apply logarithm on both sides:} \\ \log (a^x)=\log (b) \\ \text{ by the exponent rule of logarithms} \\ x\log (a)=\log (b) \end{gathered}$

For this problem we have:

$\begin{gathered} \log \frac{A}{P}=\log ((1+r)^t) \\ \log \frac{A}{P}=t\cdot\log (1+r) \end{gathered}$

Now we have to divide both sides by log(1+r) to clear t:

$\begin{gathered} \frac{\log\frac{A}{P}}{\log(1+r)}=t\cdot\frac{\log (1+r)}{\log (1+r)} \\ t=\frac{\log\frac{A}{P}}{\log(1+r)} \end{gathered}$

And finally we just have to replace the values into this equation we found: A = 11500, P = 6500 and r = 0.035:

$\begin{gathered} t=\frac{\log \frac{11500}{6500}}{\log (1+0.035)} \\ t=\frac{\log \frac{23}{13}}{\log 1.035} \\ t\approx16.6 \end{gathered}$