Question

find the amount in a continuously compounded account for the following condition. principal, $3000; annual interest rate, 5.4%; time, 5 years the balance after 5 years is $\square. (round the final answer to the nearest cent as needed. round all intermediate values to five decimal places as needed.)

Explanation:

Step1: Recall the formula for continuous compounding

The formula for the amount \( A \) in a continuously compounded account is \( A = Pe^{rt} \), where \( P \) is the principal, \( r \) is the annual interest rate (in decimal form), \( t \) is the time in years, and \( e \) is the base of the natural logarithm (approximately \( 2.71828 \)).

Step2: Convert the interest rate to decimal

The annual interest rate is \( 5.4\% \). To convert this to a decimal, we divide by \( 100 \): \( r=\frac{5.4}{100} = 0.054 \).

Step3: Identify the values of \( P \), \( r \), and \( t \)

We are given that \( P=\$3000 \), \( r = 0.054 \), and \( t = 5 \) years.

Step4: Substitute the values into the formula

Substitute \( P = 3000 \), \( r=0.054 \), and \( t = 5 \) into the formula \( A=Pe^{rt} \):
\[
A=3000\times e^{0.054\times5}
\]

Step5: Calculate the exponent

First, calculate the exponent \( rt \): \( 0.054\times5=0.27 \).

Step6: Calculate \( e^{0.27} \)

Using a calculator, \( e^{0.27}\approx1.310027 \) (rounded to six decimal places).

Step7: Calculate the amount \( A \)

Multiply \( 3000 \) by \( e^{0.27} \): \( A = 3000\times1.310027\approx3930.08 \) (rounded to the nearest cent).

Answer:

\( 3930.08 \)