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4100 dollars is placed in an account with an annual interest rate of
6.5%. To the nearest tenth of a year, how long will it take for the account value to reach 18400 dollars?
Answer:

$4100$ dollars is placed in an account with an annual interest rate of $6.5 \%$ . To the nearest tenth of a year, how long will it take for the account value to reach $18400$ dollars? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q.

4100

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

6.5 \%

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

18400

dollars?

\newline

Answer:

Identify Interest Type: Determine the type of interest being applied. $\newline$ Since the problem does not specify compound or simple interest, we will assume compound interest, which is more common in savings accounts.
Compound Interest Formula: Identify the formula for compound interest. $\newline$ 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.
Convert Interest Rate: Convert the annual interest rate from a percentage to a decimal. $6.5\%$ as a decimal is $0.065$ .
Assume Compounding Frequency: Assume that the interest is compounded once per year $n=1$ for simplicity, as the problem does not specify. $\newline$ This means the formula simplifies to $A = P(1 + r)^t$ .
Set Up and Solve Equation: Set up the equation with the given values and solve for $t$ . $18400 = 4100(1 + 0.065)^t$
Isolate Exponential Part: Divide both sides by $\$4100$ to isolate the exponential part of the equation. $\newline$ $\frac{18400}{4100} = (1 + 0.065)^t$
Calculate Left Side: Calculate the left side of the equation. $\newline$ $(18400 / 4100) = 4.4878$
Take Natural Logarithm: Take the natural logarithm of both sides to solve for $t$ . $\ln(4.4878) = \ln((1 + 0.065)^t)$
Use Logarithm Properties: Use the properties of logarithms to bring down the exponent. $\newline$ $\ln(4.4878) = t \times \ln(1 + 0.065)$
Calculate Natural Logarithm: Calculate the natural logarithm of $1 + 0.065$ . $\ln(1 + 0.065) = \ln(1.065)$
Perform Calculations: Perform the calculations. $\newline$ $\ln(4.4878) \approx 1.5007$ $\newline$ $\ln(1.065) \approx 0.063014$
Find t Value: Divide the natural logarithm of $4.4878$ by the natural logarithm of $1.065$ to find $t$ . $\newline$ t = rac{1.5007}{0.063014}
Calculate Final Answer: Calculate the value of $t$ . $t \approx 23.82$
Round to Nearest Tenth: Round the answer to the nearest tenth of a year. $\newline$ $t \approx 23.8$ years

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