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Question A person places \( \$ 26200 \) in an investment account earning an annual rate of \( 3.2 \% \), compounded continuously, Using the formula \( V=P e^{r t} \), where \( V \) is the value of the account in \( t \) years, \( P \) is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, in the account after 2o years.

Ask by Parry Brooks. in the United States
Mar 21,2025

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Answer

The amount of money in the account after 20 years is approximately \$49,687.80.

Solution

To determine the amount of money in the account after 20 years, we can use the formula for continuous compounding: \[ V = P e^{r t} \] Given: - \( P = \$26200 \) (initial investment) - \( r = 3.2\% = 0.032 \) (annual rate of interest) - \( t = 20 \) years Substitute the given values into the formula: \[ V = 26200 \times e^{0.032 \times 20} \] Now, we can calculate the value of \( V \) using the formula. Calculate the value by following steps: - step0: Calculate: \(26200e^{0.032\times 20}\) - step1: Multiply the numbers: \(26200e^{0.64}\) - step2: Convert the expressions: \(26200e^{\frac{16}{25}}\) - step3: Rewrite the expression: \(26200\sqrt[25]{e^{16}}\) The amount of money in the account after 20 years is approximately $49687.80 to the nearest cent.

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To find out how much money will be in the account after 20 years, we can use the formula \( V = P e^{rt} \). Here, \( P = 26200 \), \( r = 0.032 \) (which is 3.2% expressed as a decimal), and \( t = 20 \). Plugging in the values: \[ V = 26200 \cdot e^{(0.032 \cdot 20)} \] First, calculate \( 0.032 \cdot 20 = 0.64 \), so: \[ V = 26200 \cdot e^{0.64} \] Calculating \( e^{0.64} \approx 1.85733 \): \[ V \approx 26200 \cdot 1.85733 \approx 48629.88 \] Thus, the amount of money in the account after 20 years, to the nearest cent, is approximately **$48,629.88**. Time to celebrate your future wealth! This compound interest magic shows just how powerful saving can be over time, turning even a modest initial investment into a tidy sum. Plus, when you invest early, you give your money the gift of time to grow – and who wouldn’t want to see their cash blossom like spring flowers? As you ponder your investment strategy, consider diversifying your portfolio across different asset classes like stocks, bonds, or real estate. This can reduce risk and enhance potential returns. Just remember, the market is like a roller coaster – thrilling and sometimes dizzying – so hold on tight and enjoy the ride!

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