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d the time for an investment to double at the given annual interest rate

9.5%

◻ years
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◻
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MY NOTES
LARCPALGFL2 11.6.002.
Find the annual interest rate. (Round your answer to the nearest whole number.)

Principal
Balance
Time
Compounding

$5000

$34,070.12
15 years
Quarterly

◻

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My Assignment

d the time for an investment to double at the given annual interest rate $\newline$ $9.5 \%$ $\newline$ $\square$ years $\newline$ Need Help? $\square$ $\newline$ Submit Answer $\newline$ [ $0$ / $20$ Points] $\newline$ DETAILS $\newline$ MY NOTES $\newline$ LARCPALGFL $2$ $11$ . $6$ . $002$ . $\newline$ Find the annual interest rate. (Round your answer to the nearest whole number.) $\newline$ \begin{tabular}{|c|c|c|c|} $\newline$ \hline Principal & Balance & Time & Compounding \\ $\newline$ \hline $\$ 5000$ & $\$ 34,070.12$ & $15$ years & Quarterly \\ $\newline$ \hline $\square$ & & & \\ $\newline$ \hline d Help? & Read It & & \\ $\newline$ \hline $\newline$ \end{tabular} $\newline$ Submit Answer $\newline$ Home $\newline$ My Assignment

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Q. d the time for an investment to double at the given annual interest rate

\newline

9.5 \%

\newline

\square

years

\newline

Need Help?

\square

\newline

Submit Answer

\newline

[

0

20

Points]

\newline

DETAILS

\newline

MY NOTES

\newline

LARCPALGFL

2

11

6

002

\newline

Find the annual interest rate. (Round your answer to the nearest whole number.)

\newline

\begin{tabular}{|c|c|c|c|}

\newline

\hline Principal & Balance & Time & Compounding \\

\newline

\hline

\$ 5000

\$ 34,070.12

15

years & Quarterly \\

\newline

\hline

\square

& & & \\

\newline

\hline d Help? & Read It & & \\

\newline

\hline

\newline

\end{tabular}

\newline

Submit Answer

\newline

Home

\newline

My Assignment

Divide and Calculate: First, we need to divide $72$ by the annual interest rate of $9.5\%$ to find the approximate time in years it will take for the investment to double. $\newline$ Time to double $\approx \frac{72}{9.5}$
Rounding Time: Performing the calculation: $\newline$ Time to double $\approx \frac{72}{9.5} \approx 7.57894736842$
Given Values and Find Rate: Since we need to round the answer to the nearest whole number, we round $7.57894736842$ to $8$ years.
Rearrange and Solve for r: We are given the following values: $\newline$ Principal $P = \$5000$ $\newline$ Balance $A = \$34,070.12$ $\newline$ Time $t = 15$ years $\newline$ Compounding frequency $n = 4$ times a year $\newline$ We need to find the annual interest rate $r$ .
Isolate Compound Interest Factor: We can rearrange the compound interest formula to solve for $r$ : $A = P(1 + \frac{r}{n})^{nt}$ $34,070.12 = 5000(1 + \frac{r}{4})^{4 \times 15}$
Take $60$ th Root: Divide both sides by the principal to isolate the compound interest factor: $\newline$ $\frac{34,070.12}{5000} = (1 + \frac{r}{4})^{4\times15}$ $\newline$ $6.814024 = (1 + \frac{r}{4})^{60}$
Calculate Root: Take the $60^{\text{th}}$ root of both sides to solve for $(1 + r/4)$ : $\newline$ $(6.814024)^{\frac{1}{60}} = 1 + \frac{r}{4}$
Subtract and Solve for $r$ : Calculate the $60^{\text{th}}$ root of $6.814024$ : $\left(6.814024\right)^{\frac{1}{60}} \approx 1.029993$
Multiply and Find r: Subtract $1$ from both sides to solve for $r/4$ : $\newline$ $1.029993 - 1 = \frac{r}{4}$ $\newline$ $0.029993 = \frac{r}{4}$
Convert to Percentage: Multiply both sides by $4$ to solve for $r$ : $0.029993 \times 4 = r$ $0.119972 = r$
Round Interest Rate: Convert the decimal to a percentage to find the annual interest rate: $\newline$ $r = 0.119972 \times 100\%$ $\newline$ $r \approx 11.9972\%$
Round Interest Rate: Convert the decimal to a percentage to find the annual interest rate: $\newline$ $r = 0.119972 \times 100\%$ $\newline$ $r \approx 11.9972\%$ Round the annual interest rate to the nearest whole number: $\newline$ $r \approx 12\%$