Question

suppose that ( p ) dollars in principal is invested for ( t ) years at the given interest rates with continuous compounding. determine the amount that the investment is worth at the end of the given time period.
( p = $ 8000, t = 13 mathrm{yr} )
(a) ( 3 % ) interest
(b) ( 4 % ) interest
(c) ( 4.5 % ) interest
part: ( 0 / 3 )
part 1 of 3
(a) at ( 3 % ) interest rate, the investment will be worth ( $ ) at the end of ( 13 mathrm{yr} ).

Explanation:

Step1: Recall the formula for continuous compounding

The formula for continuous compounding is \(A = Pe^{rt}\), where \(A\) is the amount of money accumulated after \(n\) years, including interest, \(P\) is the principal amount (the initial amount of money), \(r\) is the annual interest rate (in decimal form), and \(t\) is the time the money is invested for in years.

Step2: Convert the interest rate to decimal form

For \(3\%\) interest, \(r = 0.03\). Given \(P=\$8000\) and \(t = 13\) years.

Step3: Substitute the values into the formula

Substitute \(P = 8000\), \(r=0.03\), and \(t = 13\) into \(A=Pe^{rt}\). So \(A = 8000\times e^{0.03\times13}\).

Step4: Calculate the exponent

First, calculate \(0.03\times13=0.39\). Then \(A = 8000\times e^{0.39}\).

Step5: Evaluate \(e^{0.39}\)

Using a calculator, \(e^{0.39}\approx1.478619\).

Step6: Calculate the value of \(A\)

Multiply \(8000\times1.478619\approx11728.95\).

Answer:

\(11728.95\)