Math / Calculus

QuestionQuestion 12
You deposit $\$ 5000$ in an account earning $7 \%$ interest compounded continuously. How much will you have in the account in 10 years? \\square$ Next Question

Studdy Solution

STEP 1

1. The initial deposit (principal) is $\$5000$ .
2. The interest rate is $7\%$ per annum.
3. The interest is compounded continuously.
4. The time period is $10$ years.

STEP 2

1. Recall the formula for continuous compounding.
2. Substitute the given values into the formula.
3. Calculate the future value of the investment.

STEP 3

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

STEP 4

Substitute the given values into the formula:
$P = 5000, \quad r = 0.07, \quad t = 10$
$A = 5000 \times e^{0.07 \times 10}$

STEP 5

Calculate the future value:
$A = 5000 \times e^{0.7}$
Using a calculator, find $e^{0.7} \approx 2.01375$ .
$A = 5000 \times 2.01375$
$A \approx 10068.75$
The amount in the account after 10 years will be approximately:
$\boxed{10068.75}$

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