Question

32 multiple choice: 2 points
use the exponential growth model to calculate the amount of money you will have in the bank after 12 years if you deposit $32,000 into an account that pays 2.8% interest compounded continuously.

Explanation:

Step1: Recall the continuous - compounding formula

The formula for continuous compounding is \(A = Pe^{rt}\), where \(P\) is the principal amount, \(r\) is the annual interest rate (in decimal form), and \(t\) is the time in years.
Given \(P=\$32000\), \(r = 0.028\) (since \(2.8\%=0.028\)), and \(t = 12\) years.

Step2: Substitute the values into the formula

Substitute \(P = 32000\), \(r=0.028\), and \(t = 12\) into \(A=Pe^{rt}\).
We get \(A = 32000\times e^{0.028\times12}\).
First, calculate the exponent: \(0.028\times12=0.336\).
Then, find \(e^{0.336}\approx1.399\) (using a calculator, where \(e\approx2.71828\)).
Now, \(A = 32000\times1.399\).

Step3: Calculate the final amount

\(A=32000\times1.399 = 44768\).

Answer:

\(\$44768\)