The value of a car $$ (\mathrm{V}(\mathrm{t})) $$ during the first 10 years after purchase is given by $$ V(\mathrm{t})=15000 \mathrm{e}^{-.05 \mathrm{t}} $$ $$ \quad(\mathrm{t}= $$ number of years $$ ) $$ What is the average value of the car for the first 10 years $$ [0,10] $$ ? Average Value $$ =\$[?] $$ Round your answer to the nearest dollar.

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We want to find the average value of the car over the time interval $$[0,10]$$. The average value of a function $$ V(t) $$ on the interval $$[a,b]$$ is given by

$$
\text{Average} = \frac{1}{b-a} \int_a^b V(t) \, dt.
$$

Given $$ V(t) = 15000 e^{-0.05t} $$, $$ a=0 $$, and $$ b=10 $$, the average value is

$$
\text{Average} = \frac{1}{10-0} \int_0^{10} 15000 e^{-0.05t} \, dt.
$$

First, factor out the constant:

$$
\text{Average} = \frac{15000}{10} \int_0^{10} e^{-0.05t} \, dt = 1500 \int_0^{10} e^{-0.05t} \, dt.
$$

Next, compute the integral:

$$
\int e^{-0.05t} \, dt = \frac{-1}{0.05} e^{-0.05t} = -20 e^{-0.05t}.
$$

Now evaluate from $$0$$ to $$10$$:

$$
\int_0^{10} e^{-0.05t} \, dt = \left[-20 e^{-0.05t}\right]_0^{10} = \left(-20 e^{-0.5}\right) - \left(-20 e^0\right) = -20 e^{-0.5} + 20.
$$

This simplifies to

$$
\int_0^{10} e^{-0.05t} \, dt = 20 \left( 1 - e^{-0.5} \right).
$$

Substitute this back into the expression for the average:

$$
\text{Average} = 1500 \times 20 \left( 1 - e^{-0.5} \right) = 30000 \left( 1 - e^{-0.5} \right).
$$

Now, use $$ e^{-0.5} \approx 0.60653 $$:

$$
1 - e^{-0.5} \approx 1 - 0.60653 = 0.39347.
$$

Thus,

$$
\text{Average} \approx 30000 \times 0.39347 = 11804.1.
$$

Rounding to the nearest dollar, the average value of the car for the first 10 years is

$$
\boxed{\$11804}.
$$
The average value of the car over the first 10 years is approximately \$11,804.

The value of a car during the first
10 years after purchase is given by

number of years
What is the average value of the car
for the first 10 years ?
Average Value
Round your answer to the nearest dollar.

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The value of a car during the first 10 years after purchase is given by number of years What is the average value of the car for the first 10 years ? Average Value Round your answer to the nearest dollar.