Example 2
Completion requirements
| Example 2 |
Analeise’s parents want to have saved \(\$20\thinspace 000.00\) for her post secondary education by the time she graduates from high school at age 18. They started an RESP for her when she was born with \(\$10\thinspace 000.00\). The equation to determine the amount of money in the RESP by age 18 is \(20\thinspace 000 = 10\thinspace 000 (r + 1)^{18}\). What interest rate, \(r\), do they need in order to achieve their goal? Round to the nearest tenth of a percent.
Step 1: Isolate the power.
\(\begin{align}
20\thinspace 000 &= 10\thinspace 000\left( {r + 1} \right)^{18} \\
2 &= \left( {r + 1} \right)^{18} \\
\end{align}\)
Step 2: Take the \(18\)th root of each side.
\(\begin{align}
2 &= \left( {r + 1} \right)^{18} \\
\pm \sqrt[{18}]{2} &= \sqrt[{18}]{{\left( {r + 1} \right)^{18} }} \\
\pm 1.039... &= r + 1 \\
-1 \pm 1.039... &= r \\
r &= -2.039... {\rm{ and }}\thinspace r = 0.039... \\
\end{align}\)
Because \(r = -2.039... \doteq -204\%\), it can be excluded as a suitable solution value for this context.
Verify \(r = 0.039\).
The interest rate will need to be at least \(3.9\%\).
\(\begin{align}
20\thinspace 000 &= 10\thinspace 000\left( {r + 1} \right)^{18} \\
2 &= \left( {r + 1} \right)^{18} \\
\end{align}\)
Step 2: Take the \(18\)th root of each side.
\(\begin{align}
2 &= \left( {r + 1} \right)^{18} \\
\pm \sqrt[{18}]{2} &= \sqrt[{18}]{{\left( {r + 1} \right)^{18} }} \\
\pm 1.039... &= r + 1 \\
-1 \pm 1.039... &= r \\
r &= -2.039... {\rm{ and }}\thinspace r = 0.039... \\
\end{align}\)
Because \(r = -2.039... \doteq -204\%\), it can be excluded as a suitable solution value for this context.
Verify \(r = 0.039\).
| Left Side | Right Side |
|---|---|
| \(20\thinspace 000\) | \(\begin{align} &10\thinspace 000\left( {r + 1} \right)^{18} \\ &10\thinspace 000\left( {0.039... + 1} \right)^{18} \\ &10\thinspace 000\left( {1.039...} \right)^{18} \\ &19\thinspace 999.999... \\ &20\thinspace 000 \\ \end{align}\) |
| LS = RS | |
The interest rate will need to be at least \(3.9\%\).