Question

if an initial amount of $1700 is invested in an account at an annual interest rate of 2.9%, compounded continuously, the total amount in the account after t years is given by: $a(t)=1700e^{0.029t}$. find the rate of change of the total amount in the account after 4 years. (round the answer to the nearest cent.) the total amount in the account is ? by ? after 4 years.

Explanation:

Step1: Differentiate the function

The derivative of $A(t)=1700e^{0.029t}$ with respect to $t$ using the rule $\frac{d}{dt}(e^{at}) = ae^{at}$ is $A^\prime(t)=1700\times0.029e^{0.029t}=49.3e^{0.029t}$.

Step2: Evaluate at $t = 4$

Substitute $t = 4$ into $A^\prime(t)$. So $A^\prime(4)=49.3e^{0.029\times4}$.
First, calculate $0.029\times4 = 0.116$. Then $e^{0.116}\approx1.12297$.
Multiply by 49.3: $A^\prime(4)=49.3\times1.12297\approx55.36$.

Answer:

The total amount in the account is changing by $\$55.36$ after 4 years.