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Math  /  Calculus

QuestionMike deposited $10000\$ 10000 in a saving account in which interest is compounded continuously. The annual rate of interest is 4.5%4.5 \%. (a.) How much does he have in this account after 15 years? Round your answer to the nearest cent.
Amount after 15 years ≈$\approx \$ □\square

Studdy Solution

STEP 1

1. The principal amount deposited is $10,000 \$10,000 .
2. The interest is compounded continuously.
3. The annual interest rate is 4.5% 4.5\% .
4. The time period is 15 15 years.

STEP 2

1. Recall the formula for continuous compounding.
2. Substitute the given values into the formula.
3. Calculate the amount after 15 years.
4. Round the result to the nearest cent.

STEP 3

Recall the formula for continuous compounding:
A=Pert A = Pe^{rt}
where: - A A is the amount of money accumulated after n n years, including interest. - P P is the principal amount (initial deposit). - r r is the annual interest rate (decimal). - t t is the time the money is invested for in years. - e e is the base of the natural logarithm, approximately equal to 2.71828 2.71828 .

STEP 4

Substitute the given values into the formula:
P=10000,r=0.045,t=15 P = 10000, \quad r = 0.045, \quad t = 15
A=10000×e0.045×15 A = 10000 \times e^{0.045 \times 15}

STEP 5

Calculate the amount after 15 years:
A=10000×e0.675 A = 10000 \times e^{0.675}
Using a calculator, compute e0.675≈1.96403 e^{0.675} \approx 1.96403 .
A≈10000×1.96403 A \approx 10000 \times 1.96403 A≈19640.30 A \approx 19640.30

STEP 6

Round the result to the nearest cent:
Amount after 15 years ≈$19640.30\approx \$19640.30
The final amount in the account after 15 years is approximately:
19640.30 \boxed{19640.30}

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