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7600 dollars is placed in an account with an annual interest rate of
7.5%. To the nearest tenth of a year, how long will it take for the account value to reach 34700 dollars?
Answer:

$7600$ dollars is placed in an account with an annual interest rate of $7.5 \%$ . To the nearest tenth of a year, how long will it take for the account value to reach $34700$ dollars? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q.

7600

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

7.5 \%

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

34700

dollars?

\newline

Answer:

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. $\newline$ Since the problem does not specify how often the interest is compounded, we will assume it is compounded annually, so $n = 1$ .
Set up equation with given values: Set up the equation with the given values. $\newline$ We need to find $t$ when $A = 34700$ , $P = 7600$ , $r = 7.5\%$ (or $0.075$ as a decimal), and $n = 1$ . $\newline$ $34700 = 7600(1 + 0.075/1)^{(1*t)}$
Simplify the equation: Simplify the equation. $\newline$ $34700 = 7600(1 + 0.075)^{t}$ $\newline$ $34700 = 7600(1.075)^{t}$
Divide to isolate exponential part: Divide both sides by $7600$ to isolate the exponential part of the equation. $\newline$ $\frac{34700}{7600} = (1.075)^{(t)}$ $\newline$ $4.56578947368 \approx (1.075)^{(t)}$
Take natural logarithm to solve: Take the natural logarithm of both sides to solve for $t$ . $\ln(4.56578947368) = \ln((1.075)^{t})$ $\ln(4.56578947368) = t \cdot \ln(1.075)$
Divide to solve for $t$ : Divide both sides by $\ln(1.075)$ to solve for $t$ . $\newline$ $t = \frac{\ln(4.56578947368)}{\ln(1.075)}$
Calculate $t$ using calculator: Calculate the value of $t$ using a calculator. $\newline$ $t \approx \ln(4.56578947368) / \ln(1.075)$ $\newline$ $t \approx 17.6727$
Round answer to nearest tenth: Round the answer to the nearest tenth of a year. $t \approx 17.7$ years

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