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. $45,068.09 $44,778.85 $44,941.68 $44,663.23

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), \(t\) is the time in years, and \(A\) is the amount of money in the account after \(t\) 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 \(e^{0.336}\approx1.3997\) (using a calculator, where \(e\approx2.71828\)).

Step3: Calculate the value of \(A\)

Now, \(A = 32000\times1.3997\).
\(A=32000\times1.3997 = 44790.4\approx44778.85\) (due to more precise calculation of \(e^{0.336}\) when using a calculator's \(e^x\) function).

Answer:

\(\$44,778.85\)