Questions: Question 14, 5.1.25 HW Score: 46%, 11.5 of 25 points Points: 0 of 1 Find the compound amount and the amount of interest earned by the following deposit. 5900 at 3.2% compounded continuously for 8 years. The future value after 8 years is approximately (Do not round until the final answer. Then round to the nearest cent as needed.)

Question 14, 5.1.25 HW Score: 46%, 11.5 of 25 points Points: 0 of 1 Find the compound amount and the amount of interest earned by the following deposit. 5900 at 3.2% compounded continuously for 8 years. The future value after 8 years is approximately (Do not round until the final answer. Then round to the nearest cent as needed.)
Transcript text: Question 14, 5.1.25 HW Score: $46 \%, 11.5$ of 25 points Points: 0 of 1 Find the compound amount and the amount of interest earned by the following deposit. $\$ 5900$ at 3.2\% compounded continuously for 8 years. The future value after 8 years is approximately $\$$ $\square$ (Do not round until the final answer. Then round to the nearest cent as needed.)
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Solution

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Solution Steps

Step 1: Convert the Annual Interest Rate to a Decimal

The given annual interest rate is 3.2%. To convert it to a decimal, we divide by 100: \[r = \frac{3.2}{100} = 0.032\]

Step 2: Substitute the Values into the Formula

Substitute the values of \(P = 5900\), \(r = 0.032\), and \(t = 8\) years into the formula \(A = Pe^{rt}\): \[A = 5900e^{0.032 \times 8}\]

Step 3: Calculate the Compound Amount

Calculate the exponent \(rt\) and then raise \(e\) to this power, and multiply the result by \(P\): \[A = 5900 \times e^{0.256} = 7621.34\]

Step 4: Calculate the Amount of Interest Earned

The amount of interest earned is the compound amount minus the initial deposit: \[\text{Interest Earned} = A - P = 7621.34 - 5900 = 1721.34\]

Final Answer:

The compound amount after 8 years is \(\textbf{$7621.34$}\), and the interest earned is \(\textbf{$1721.34$}\).

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