Questions: How much money has to be invested at 5.1% interest compounded continuously to have 17,000 after 14 years? A. 8337.19 B. 8324.59 C. 8399.44 D. 8362.24

How much money has to be invested at 5.1% interest compounded continuously to have 17,000 after 14 years? A. 8337.19 B. 8324.59 C. 8399.44 D. 8362.24
Transcript text: How much money has to be invested at $5.1 \%$ interest compounded continuously to have $\$ 17,000$ after 14 years? A. $\$ 8337.19$ B. $\$ 8324.59$ C. $\$ 8399.44$ D. $\$ 8362.24$
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Solution

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Solution Steps

Step 1: Identify the Variables

We are given the following values:

  • \( A = 17000 \) (the amount of money after 14 years)
  • \( r = 0.051 \) (the annual interest rate)
  • \( t = 14 \) (the time in years)
Step 2: Use the Continuous Compound Interest Formula

We will use the formula for continuous compound interest: \[ A = Pe^{rt} \] To find the principal amount \( P \), we rearrange the formula: \[ P = \frac{A}{e^{rt}} \]

Step 3: Calculate the Principal Amount

Substituting the known values into the equation: \[ P = \frac{17000}{e^{0.051 \times 14}} \] Calculating this gives: \[ P \approx 8324.5863 \]

Final Answer

The amount that needs to be invested is approximately \\(\boxed{P = 8324.59}\\).

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