Question

the value of a car after it is sold decreases according to the function $v(t)=38000 - 0.7e^{t}$, where $t$ is the number of years since the car was purchased.
a. what is the value of the car after 4 years? (round to the nearest dollar.)
the value of the car after 4 years is

b. how is the value of the car changing after 4 years? (round to the nearest dollar.)
the value of the car is by

Explanation:

Step1: Find value of car after 4 years

Substitute $t = 4$ into $V(t)=38000 - 0.7e^{t}$.
$V(4)=38000-0.7e^{4}$
$V(4)=38000 - 0.7\times54.59815$
$V(4)=38000 - 38.218705\approx37962$

Step2: Find rate of change of car's value

First, find the derivative of $V(t)$. The derivative of $V(t)=38000 - 0.7e^{t}$ is $V^\prime(t)=- 0.7e^{t}$.
Then substitute $t = 4$ into $V^\prime(t)$.
$V^\prime(4)=-0.7e^{4}=-0.7\times54.59815\approx - 38$

Answer:

a. 37962
b. decreasing; 38