if an initial amount of $4400 is invested in an account at an annual interest rate of 4.4%, compounded…
if an initial amount of $4400 is invested in an account at an annual interest rate of 4.4%, compounded continuously, the total amount in the account after t years is given by: $a(t)=4400e^{0.044t}$. 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.
Answer
Explanation:
Step1: Differentiate the function
The function is $A(t) = 4400e^{0.044t}$. Using the chain - rule, if $y = e^{u}$ and $u = 0.044t$, then $\frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt}$. The derivative of $e^{u}$ with respect to $u$ is $e^{u}$, and the derivative of $u = 0.044t$ with respect to $t$ is $0.044$. So, $A^\prime(t)=4400\times0.044e^{0.044t}=193.6e^{0.044t}$.
Step2: Evaluate the derivative at $t = 4$
Substitute $t = 4$ into $A^\prime(t)$. We have $A^\prime(4)=193.6e^{0.044\times4}=193.6e^{0.176}$. Since $e^{0.176}\approx1.1923$, then $A^\prime(4)=193.6\times1.1923 = 193.6\times1.1923=230.83$.
Answer:
The total amount in the account is increasing by $$230.83$ after 4 years.