Euler's Number, e Practice

Question

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Question
Eli received $10,000 in gift money when he graduated from high school. He deposits the money into an account with a 5 percent annual interest rate, compounded continuously. Use the formula P(t)=10,000e0.05t
to determine how much money will be in his account in four years. Include cents, if needed.

Answer 1

To find out how much money Eli will have in his account after four years with continuous compounding, we will use the formula:

\[ P(t) = P_0 e^{rt} \]

where:

• \( P_0 = 10,000 \) (the initial amount),
• \( r = 0.05 \) (the annual interest rate),
• \( t = 4 \) (the time in years),
• \( e \) is Euler's number (approximately 2.71828).

Now, plug the values into the formula:

\[ P(4) = 10,000 e^{0.05 \cdot 4} \] \[ P(4) = 10,000 e^{0.20} \]

Now, calculate \( e^{0.20} \):

Using a calculator, \( e^{0.20} \) is approximately 1.2214.

Now calculate \( P(4) \):

\[ P(4) = 10,000 \times 1.2214 \approx 12,214.00 \]

Thus, the amount in his account after four years will be approximately $12,214.00.