Question

an initial amount of $1800 is invested in an account at an interest rate of 7% per year, compounded continuously. assuming that no withdrawals are made, find the amount in the account after six years. do not round any intermediate computations, and round your answer to the nearest cent.

Explanation:

Step1: Recall the formula for continuous compounding

The formula for continuous compounding is $A = Pe^{rt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $t$ is the time in years, and $e$ is the base of the natural logarithm.
Here, $P = 1800$, $r = 0.07$ (since 7% = 0.07), and $t = 6$.

Step2: Substitute the values into the formula

Substitute $P = 1800$, $r = 0.07$, and $t = 6$ into the formula:
$A = 1800 \times e^{(0.07 \times 6)}$

Step3: Calculate the exponent

First, calculate the exponent: $0.07 \times 6 = 0.42$

Step4: Calculate the value of $e^{0.42}$

Using a calculator, $e^{0.42} \approx 1.521961556$

Step5: Calculate the final amount

Multiply 1800 by 1.521961556:
$A = 1800 \times 1.521961556 \approx 2739.530801$

Answer:

$\$2739.53$