Bobby and Emilia deposit $600.00 into a savings account which earns 3% interest compounded continuously. They want to use the money in the account to go on a trip in 1 year. How much will they be able to spend?

Question

Use the formula A=Pert, where A is the balance (final amount), P is the principal (starting amount), e is the base of natural logarithms (≈2.71828), r is the interest rate expressed as a decimal, and t is the time in years.
Round your answer to the nearest cent.

Answer 1

To find the amount that Bobby and Emilia will have in their savings account after one year with continuous compounding, we can use the formula:

\[ A = Pe^{rt} \]

Where:

• \(A\) is the amount of money accumulated after time \(t\),
• \(P\) is the principal amount (the initial deposit),
• \(e\) is approximately 2.71828,
• \(r\) is the annual interest rate (as a decimal),
• \(t\) is the time in years.

Given:

• \(P = 600.00\)
• \(r = 0.03\) (3% as a decimal)
• \(t = 1\)

Now we can plug in the values:

\[ A = 600.00 \cdot e^{0.03 \cdot 1} \]

First, calculate \(0.03 \cdot 1\):

\[ 0.03 \cdot 1 = 0.03 \]

Now calculate \(e^{0.03}\):

Using the approximation \(e \approx 2.71828\):

\[ e^{0.03} \approx 2.71828^{0.03} \approx 1.03045453395 \]

Now, multiply the principal by \(e^{0.03}\):

\[ A \approx 600.00 \cdot 1.03045453395 \]

Calculating the multiplication:

\[ A \approx 618.27272037 \]

Now round to the nearest cent:

\[ A \approx 618.27 \]

Thus, Bobby and Emilia will be able to spend approximately $618.27.