Question

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 $5,000in an account that is compounded continuously as an annual interest rate of 2.5%. In two 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 solve the problem using the formula \( A = Pe^{rt} \), we will follow these steps:

1. Identify the variables:

   • \( P = 5000 \) (the principal amount)
   • \( r = 0.025 \) (the annual interest rate as a decimal)
   • \( t = 2 \) (the time in years)

2. Substitute the variables into the formula: \[ A = Pe^{rt} \] Now substituting the known values: \[ A = 5000 \cdot e^{(0.025 \cdot 2)} \]

3. Calculate the exponent: First calculate \( rt \): \[ rt = 0.025 \cdot 2 = 0.05 \] So now the equation looks like: \[ A = 5000 \cdot e^{0.05} \]

4. Calculate \( e^{0.05} \): Using a calculator to find \( e^{0.05} \): \[ e^{0.05} \approx 1.051271 \]

5. Substitute back into the equation: \[ A = 5000 \cdot 1.051271 \]

6. Perform the multiplication: \[ A \approx 5000 \cdot 1.051271 \approx 5256.355 \]

7. Round the result to the nearest dollar: \[ A \approx 5256 \]

Thus, Dora's investment will be worth approximately $5,256 after 2 years.