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Suppose you invest
$10000 at
5.1% annual interest, compounded weekly. How long will it take to double your money? Round to the nearest year. (Hint this will be a whole number)

Suppose you invest $\$ 10000$ at $5.1 \%$ annual interest, compounded weekly. How long will it take to double your money? Round to the nearest year. (Hint this will be a whole number)

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Exponential growth and decay: word problems

Full solution

Q. Suppose you invest

\$ 10000

at

5.1 \%

annual interest, compounded weekly. How long will it take to double your money? Round to the nearest year. (Hint this will be a whole number)

Identify formula for compound interest: Identify the formula to use for compound interest. $\newline$ The formula for compound interest is $A = P(1 + \frac{r}{n})^{nt}$ , where: $\newline$ $A$ is the amount of money accumulated after $n$ years, including interest. $\newline$ $P$ is the principal amount (the initial amount of money). $\newline$ $r$ is the annual interest rate (decimal). $\newline$ $n$ is the number of times that interest is compounded per year. $\newline$ $t$ is the time the money is invested for in years. $\newline$ We want to find $t$ when $A$ is double the principal $P$ .
Set up equation with given values: Set up the equation with the given values. $\newline$ We know that $A = 2P$ (since we want to double the money), $P = \$10,000$ , $r = 5.1\%$ or $0.051$ (as a decimal), and $n = 52$ (since interest is compounded weekly). $\newline$ So, the equation becomes $2P = P(1 + 0.051/52)^{52t}$ .
Simplify the equation: Simplify the equation. $\newline$ We can divide both sides by $P$ to get $2 = (1 + 0.051/52)^{52t}$ .
Solve for t: Solve for t. $\newline$ To solve for $t$ , we need to take the natural logarithm (ln) of both sides: $\newline$ $\ln(2) = \ln((1 + 0.051/52)^{52t})$ . $\newline$ Using the power rule of logarithms, we get: $\newline$ $\ln(2) = 52t \cdot \ln(1 + 0.051/52)$ .
Isolate t: Isolate t. $\newline$ Divide both sides by $52 \times \ln(1 + 0.051/52)$ to get: $\newline$ $t = \frac{\ln(2)}{52 \times \ln(1 + 0.051/52)}$ .
Calculate value of t: Calculate the value of t. $\newline$ Using a calculator, we find: $\newline$ $t \approx \frac{\ln(2)}{52 \times \ln(1 + 0.051/52)}$ . $\newline$ $t \approx \frac{0.69314718056}{52 \times \ln(1.00098039216)}$ . $\newline$ $t \approx \frac{0.69314718056}{52 \times 0.00098019802}$ . $\newline$ $t \approx \frac{0.69314718056}{0.05097029604}$ . $\newline$ $t \approx 13.60546875$ .
Round answer to nearest year: Round the answer to the nearest year. $\newline$ Since we are asked to round to the nearest year, $t \approx 14$ years.

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