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Suppose you have
$9500 deposited at
2.75% compounded semiannually. About long will it take your balanc to increase to
$10900 ?

Suppose you have $\$ 9500$ deposited at $2.75 \%$ compounded semiannually. About long will it take your balanc to increase to $\$ 10900$ ?

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Exponential growth and decay: word problems

Full solution

Q. Suppose you have

\$ 9500

deposited at

2.75 \%

compounded semiannually. About long will it take your balanc to increase to

\$ 10900

?

Identify 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.
Plug in Values: Plug in the known values into the formula. $\newline$ We know that: $\newline$ $P = \$9500$ $\newline$ $A = \$10900$ $\newline$ $r = 2.75\%$ or $0.0275$ (as a decimal) $\newline$ $n = 2$ (since the interest is compounded semiannually) $\newline$ We need to find $t$ . $\newline$ So, the equation becomes $\$10900 = \$9500(1 + 0.0275/2)^{2t}$ .
Solve for t: Solve for t. $\newline$ First, divide both sides by $\$9500$ to isolate the exponential part of the equation: $\newline$ $\$10900 / \$9500 = (1 + 0.0275/2)^{2t}$ $\newline$ $1.14736842105 = (1 + 0.01375)^{2t}$
Simplify Equation: Simplify the equation. $\newline$ $1.14736842105 = (1.01375)^{(2t)}$ $\newline$ Now we need to take the logarithm of both sides to solve for $t$ .
Apply Logarithm: Apply the logarithm to both sides. $\newline$ Using the property of logarithms that $\log(b^x) = x\log(b)$ , we can write: $\newline$ $\log(1.14736842105) = 2t \cdot \log(1.01375)$
Isolate t: Isolate $t$ . $\newline$ Divide both sides by $(2 \cdot \log(1.01375))$ to solve for $t$ : $\newline$ $t = \frac{\log(1.14736842105)}{(2 \cdot \log(1.01375))}$
Calculate $t$ : Calculate the value of $t$ . $\newline$ Using a calculator, we find: $\newline$ $t \approx \log(1.14736842105) / (2 \times \log(1.01375))$ $\newline$ $t \approx 0.0602735772 / (2 \times 0.0059564723)$ $\newline$ $t \approx 0.0602735772 / 0.0119129446$ $\newline$ $t \approx 5.0607$
Interpret Result: Interpret the result. $\newline$ Since $t$ is the number of years and we have found that $t \approx 5.0607$ , it will take approximately $5.0607$ years for the balance to increase from $\$9500$ to $\$10900$ with an annual interest rate of $2.75\%$ compounded semiannually.

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