Question

Jessica deposited $4000 into a savings account at 4.5% annual interest compounded continuously. What is her balance after 10 years?

a) $7,183.77
b) $7,750.00
c) $7,950.00
d) $8,183.77

Answer 1

The formula for compound interest compounded continuously is given by:

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

where:

- \( A \) is the final amount (the balance after \( t \) years),

- \( P \) is the principal amount (initial deposit),

- \( r \) is the annual interest rate (in decimal form),

- \( t \) is the time the money is invested or borrowed for in years, and

- \( e \) is the mathematical constant approximately equal to 2.71828.

In this case:

- \( P = $4000 \),

- \( r = 0.045 \) (4.5% as a decimal),

- \( t = 10 \) years.

\[ A = 4000 \cdot e^{0.045 \cdot 10} \]

Using a calculator, the result is approximately $7,183.77.

So, the correct answer is:

a) $7,183.77

Answer 2

Using the continuous compounding formula A = Pe^(rt), Jessica's balance after depositing $4000 in a savings account at 4.5% annual interest for 10 years is approximately $7,183.77, which is option (a).

Step-by-step explanation:

When Jessica deposits $4000 into a savings account at 4.5% annual interest compounded continuously, she uses the formula for continuous compounding, which is A = Pert. In this formula, A is the amount in the account after time t, P is the principal amount (initial deposit), r is the annual interest rate (in decimal), t is the time in years, and e is the base of the natural logarithm, approximately equal to 2.71828.

To find her balance after 10 years, we use the given values: P = $4000, r = 0.045 (4.5% as a decimal), and t = 10. Substituting these into the formula gives us:

A = $4000 × e(0.045 × 10)

After calculating the exponent and multiplication, we find that Jessica's balance after 10 years would be approximately $7,183.77. Hence, the answer is (a) $7,183.77.