Question

Use the equation A=Pe^rt to answer each question. Show all work.

Answer 1

the interest rate is 2.7 %

Explanation

the continuos compound formula states that:

$\begin{gathered} A=P*e^{rt} \\ where\text{ A }is\text{ the final amount} \\ Pis\text{ the initial amount} \\ e\text{ is a math constant} \\ \text{r is the time} \end{gathered}$

so, we can isolate r

$\begin{gathered} A=Pe^{rt} \\ divide\text{ both sides by P} \\ \frac{A}{P}=\frac{Pe^{rt}}{P} \\ \frac{A}{P}=e^{rt} \\ ln(\frac{A}{P})=ln(e^{rt}) \\ ln(\frac{A}{P})=rt \\ divide\text{ both sides by t} \\ \frac{ln(\frac{A}{P})}{t}=\frac{rt}{t} \\ r=\frac{ln(\frac{A}{P})}{t} \end{gathered}$

hence

Step 1

a) Let

$\begin{gathered} P=\text{ 8000} \\ A=13006.40 \\ t=\text{ 18} \end{gathered}$

b) now, replace in the formula to find the rate:

$\begin{gathered} r=\frac{ln(\frac{A}{P})}{t} \\ r=\frac{ln(\frac{13006.40}{8000})}{18} \\ r=\frac{0.486}{18} \\ r=0.027 \end{gathered}$

to get the percentage form, multiply by 100 %

$r=0.027*100\text{ \% =2.7 \%}$

so, the interest rate is 2.7 %

I hope this helps you