QuestionAugust 5, 2025

How much would you have to deposit in an account with a 4.75% interest rate, compounded continuously, to have 20,000 in your account 20 years later? P= [?] Round to the nearest cent.

How much would you have to deposit in an account with a 4.75% interest rate, compounded continuously, to have 20,000 in your account 20 years later? P= [?] Round to the nearest cent.
How much would you have to deposit in an account with a 4.75% interest rate, compounded continuously, to have 20,000 in your account 20 years later? P= [?] Round to the nearest cent.

Solution
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Answer

7738.99 Explanation 1. Identify the formula for continuous compounding Use the formula for continuous compounding: **A = Pe^{rt}**, where A is the amount, P is the principal, r is the rate, and t is time. 2. Rearrange the formula to solve for P Rearrange to find P: **P = \frac{A}{e^{rt}}**. 3. Substitute the given values Substitute A = 20000, r = 0.0475, and t = 20 into the formula: P = \frac{20000}{e^{0.0475 \times 20}}. 4. Calculate the exponent Compute e^{0.0475 \times 20} \approx e^{0.95}. 5. Calculate P Evaluate P = \frac{20000}{e^{0.95}} \approx \frac{20000}{2.5857}. 6. Final calculation Calculate P \approx 7738.99.

Explanation

1. Identify the formula for continuous compounding<br /> Use the formula for continuous compounding: **$A = Pe^{rt}$**, where $A$ is the amount, $P$ is the principal, $r$ is the rate, and $t$ is time.<br /><br />2. Rearrange the formula to solve for $P$<br /> Rearrange to find $P$: **$P = \frac{A}{e^{rt}}$**.<br /><br />3. Substitute the given values<br /> Substitute $A = 20000$, $r = 0.0475$, and $t = 20$ into the formula: $P = \frac{20000}{e^{0.0475 \times 20}}$.<br /><br />4. Calculate the exponent<br /> Compute $e^{0.0475 \times 20} \approx e^{0.95}$.<br /><br />5. Calculate $P$<br /> Evaluate $P = \frac{20000}{e^{0.95}} \approx \frac{20000}{2.5857}$.<br /><br />6. Final calculation<br /> Calculate $P \approx 7738.99$.
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