Apply the formula A = Pe^rt , where A is the amount accrued, P is the principal, r is the rate of interest, and t is the time, in years. Dora invests $5,000 in an account that is compounded continuously at an annual interest rate of 2.5%. In 10 years, how much will Dora's investment be worth to the nearest dollar? Show all the steps Dora must take to solve this problem. You can type your work below OR upload a picture of your hand written work.

Answer 1

To solve for the amount \( A \) that Dora's investment will be worth after 10 years using the formula for continuous compounding, \( A = Pe^{rt} \), we will follow these steps:

1. Identify the given values:

   • Principal \( P = 5000 \)
   • Annual interest rate \( r = 0.025 \) (which is 2.5% expressed as a decimal)
   • Time \( t = 10 \) years

2. Plug the values into the formula:

   \[ A = 5000 \times e^{(0.025 \times 10)} \]

3. Calculate the exponent:

   \[ 0.025 \times 10 = 0.25 \]

4. Calculate \( e^{0.25} \). Using a calculator, \( e^{0.25} \) is approximately:

   \[ e^{0.25} \approx 1.284025 \]

5. Now, substitute \( e^{0.25} \) back into the formula:

   \[ A \approx 5000 \times 1.284025 \]

6. Calculate \( 5000 \times 1.284025 \):

   \[ A \approx 6420.125 \]

7. Round to the nearest dollar:

   \[ A \approx 6420 \]

Thus, after 10 years, Dora's investment will be worth approximately $6,420.

Answer 2

Im sorry I cant copy the numbers its not letting me

Answer 3

That's okay! Let me provide you with the solution in a clear format without relying on copying numbers.

Steps to Calculate the Value of Dora's Investment

1. Identify the Variables:

   • Principal (\( P \)): $5,000
   • Annual Interest Rate (\( r \)): 2.5% (as a decimal: 0.025)
   • Time (\( t \)): 10 years

2. Use the Continuous Compounding Formula: \[ A = Pe^{rt} \] Where:

   • \( A \) = Amount accrued
   • \( P \) = Principal
   • \( e \) = Euler's number (approximately 2.71828)
   • \( r \) = Annual interest rate
   • \( t \) = Time in years

3. Plug in the Values: \[ A = 5000 \times e^{(0.025 \times 10)} \]

4. Calculate the Exponent: \[ rt = 0.025 \times 10 = 0.25 \]

5. Calculate \( e^{0.25} \): Using a calculator: \[ e^{0.25} \approx 1.284025 \]

6. Substitute Back into the Equation: \[ A \approx 5000 \times 1.284025 \]

7. Calculate the Final Amount: \[ A \approx 6420.125 \]

8. Round to the Nearest Dollar: Therefore: \[ A \approx 6420 \]

Conclusion

After 10 years, Dora's investment will be worth approximately $6,420.

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