An investment company pays 6% compounded semiannually. You want to have $17,000 in the future. (A) How much should you deposit now to have that amount 5 years from now? $◻ (Round to the nearest cent.)

Identify Formula: Identify the formula for compound interest. The formula for compound interest is A = P(1 + r/n)^(nt), where: A = the future value of the investment/loan, including interest P = the principal investment amount (the initial deposit or loan amount) r = the annual interest rate (decimal) n = the number of times that interest is compounded per year t = the time the money is invested or borrowed for, in yearsConvert Rate: Convert the annual interest rate from a percentage to a decimal. The annual interest rate is 6%, which as a decimal is 0.06.Determine Values: Determine the values of n, t, and A. Since the interest is compounded semiannually, n = 2. The time t is 5 years. The future value A we want to have is $17,000.Substitute and Solve: Substitute the values into the compound interest formula and solve for P. We have A = $17,000, r = 0.06, n = 2, and t = 5. So, $17,000 = P(1 + 0.06/2)^(2*5)Calculate Values: Calculate the value inside the parentheses and the exponent. 1 + 0.06/2 = 1 + 0.03 = 1.03 2*5 = 10 Now the equation is $17,000 = P(1.03)^10Calculate Exponent: Calculate (1.03)^10. (1.03)^10 ≈ 1.343916379Divide and Solve: Divide both sides of the equation by (1.03)^10 to solve for P. $17,000 / 1.343916379 ≈ P P ≈ $12,641.51Round Result: Round the result to the nearest cent. P ≈ $12,641.51 (rounded to the nearest cent)

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An investment company pays $6\%$ compounded semiannually. You want to have $\$17,000$ in the future. $\newline$ (A) How much should you deposit now to have that amount $5$ years from now? $\newline$ $\$$ $\square$ (Round to the nearest cent.)

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Algebra 1

Exponential growth and decay: word problems

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Q. An investment company pays

6\%

compounded semiannually. You want to have

\$17,000

in the future.

\newline

(A) How much should you deposit now to have that amount

5

years from now?

\newline

\$

\square

(Round to the nearest cent.)

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 future value of the investment/loan, including interest $\newline$ $P$ = the principal investment amount (the initial deposit or loan amount) $\newline$ $r$ = 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 or borrowed for, in years
Convert Rate: Convert the annual interest rate from a percentage to a decimal. $\newline$ The annual interest rate is $6\%$ , which as a decimal is $0.06$ .
Determine Values: Determine the values of $n$ , $t$ , and $A$ . $\newline$ Since the interest is compounded semiannually, $n = 2$ . $\newline$ The time $t$ is $5$ years. $\newline$ The future value $A$ we want to have is $\$\(17$,$000$\$).
Substitute and Solve: Substitute the values into the compound interest formula and solve for $P$. We have $A = \$17,000$, $r = 0.06$, $n = 2$, and $t = 5$. So, $\$17,000 = P(1 + \frac{0.06}{2})^{(2\cdot 5)}$
Calculate Values: Calculate the value inside the parentheses and the exponent.$\newline$$1 + \frac{0.06}{2} = 1 + 0.03 = 1.03$$\newline$$2\times5 = 10$$\newline$Now the equation is $\$17,000 = P(1.03)^{10}$
Calculate Exponent: Calculate $(1.03)^{10}$.$(1.03)^{10} \approx 1.343916379$
Divide and Solve: Divide both sides of the equation by $(1.03)^{10}$ to solve for $P$.
$\$17,000 / 1.343916379 \approx P$
P \approx \$($12$,$641$.$51$)
Round Result: Round the result to the nearest cent.$\newline$P $\approx$ $\$12,641.51$ (rounded to the nearest cent)