Apply the formula A=Pert

Question

, 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 the steps that Dora must take to solve this problem.

Answer 1

To find out how much Dora's investment will be worth after 10 years with continuous compounding, we will use the formula:

\[ A = P e^{rt} \]

where:

• \( A \) is the amount accrued (the future value of the investment),
• \( P \) is the principal (the initial amount of money),
• \( r \) is the annual interest rate (in decimal form),
• \( t \) is the time in years,
• \( e \) is the base of the natural logarithm (approximately equal to 2.71828).

Given:

• \( P = 5000 \) (the amount Dora invests),
• \( r = 2.5% = 0.025 \) (the annual interest rate in decimal),
• \( t = 10 \) years (the time period).

Step 1: Identify the values for the variables

• \( P = 5000 \)
• \( r = 0.025 \)
• \( t = 10 \)

Step 2: Plug the values into the formula

Now, substitute the values into the formula:

\[ A = 5000 \cdot e^{0.025 \cdot 10} \]

Step 3: Calculate the exponent

First, calculate \( 0.025 \cdot 10 \):

\[ 0.025 \cdot 10 = 0.25 \]

Step 4: Compute \( e^{0.25} \)

Now, calculate \( e^{0.25} \). Using a calculator, we find:

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

Step 5: Calculate \( A \)

Now, plug this value back into the equation:

\[ A = 5000 \cdot 1.284025 \]

Calculating this gives:

\[ A \approx 6420.125 \]

Step 6: Round to the nearest dollar

Finally, round \( 6420.125 \) to the nearest dollar:

\[ A \approx 6420 \]

Conclusion

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