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8700 dollars is placed in an account with an annual interest rate of
8%. To the nearest year, how long will it take for the account value to reach 20000 dollars?
Answer:

$8700$ dollars is placed in an account with an annual interest rate of $8 \%$ . To the nearest year, how long will it take for the account value to reach $20000$ dollars? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q.

8700

dollars is placed in an account with an annual interest rate of

8 \%

. To the nearest year, how long will it take for the account value to reach

20000

dollars?

\newline

Answer:

Identify Type of Interest: Identify the type of interest being applied. $\newline$ Since the problem does not specify compound or simple interest, we will assume compound interest is being applied annually.
Use Compound Interest Formula: Use the formula for compound interest to find the time. $\newline$ The formula for compound interest is $A = P(1 + r/n)^{(nt)}$ , where: $\newline$ $A$ = the amount of money accumulated after $n$ years, including interest. $\newline$ $P$ = the principal amount (the initial amount of money). $\newline$ $r$ = the annual interest rate (decimal). $\newline$ $n$ = the number of times that interest is compounded per year. $\newline$ $t$ = the time the money is invested for, in years. $\newline$ In this case, $A = \$20000$ , $P = \$8700$ , $r = 8\%$ or $A$ $0$ , and $A$ $1$ (since it's compounded annually).
Convert Rate to Decimal: Convert the percentage rate to a decimal and plug in the values. $\newline$ $A = 20000$ , $P = 8700$ , $r = 0.08$ , $n = 1$ . $\newline$ We need to solve for $t$ . $\newline$ $20000 = 8700(1 + 0.08/1)^{(1\cdot t)}$
Simplify Equation and Solve: Simplify the equation and solve for $t$ . $20000 = 8700(1 + 0.08)^t$ $20000 = 8700(1.08)^t$
Isolate Exponential Part: Divide both sides by $8700$ to isolate the exponential part. $\newline$ $\frac{20000}{8700} = (1.08)^t$ $\newline$ $2.29885057471 \approx (1.08)^t$
Take Natural Logarithm: Take the natural logarithm of both sides to solve for $t$ . $\ln(2.29885057471) = \ln((1.08)^t)$ $\ln(2.29885057471) = t \cdot \ln(1.08)$
Divide to Solve for t: Divide both sides by $\ln(1.08)$ to solve for t. $t = \frac{\ln(2.29885057471)}{\ln(1.08)}$ $t \approx 10.2447683516$
Round to Nearest Year: Round the answer to the nearest year. $\newline$ $t \approx 10$ years

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