To find the distance the car travels during the first 2 hours, we can follow these steps:
### Step-by-Step Solution
1. Understand the given velocity function:
The velocity function of the car is given by:
[tex]\[
v(t) = 13t - 1.5t^2
\][/tex]
where [tex]\( v(t) \)[/tex] is the velocity in miles per hour and [tex]\( t \)[/tex] is the time in hours.
2. Determine what we need to find:
We need to find the distance traveled by the car over the first 2 hours. To do this, we need to integrate the velocity function from [tex]\( t = 0 \)[/tex] to [tex]\( t = 2 \)[/tex].
3. Set up the integral for the distance:
The distance traveled, [tex]\( D \)[/tex], is the definite integral of the velocity function [tex]\( v(t) \)[/tex] with respect to time [tex]\( t \)[/tex] from 0 to 2:
[tex]\[
D = \int_{0}^{2} v(t) \, dt
\][/tex]
Substituting the given velocity function:
[tex]\[
D = \int_{0}^{2} (13t - 1.5t^2) \, dt
\][/tex]
4. Integrate the velocity function:
To integrate [tex]\( 13t - 1.5t^2 \)[/tex], we apply the power rule of integration:
[tex]\[
\int (13t - 1.5t^2) \, dt = \int 13t \, dt - \int 1.5t^2 \, dt
\][/tex]
[tex]\[
= 13 \int t \, dt - 1.5 \int t^2 \, dt
\][/tex]
Using the power rule [tex]\(\int t^n \, dt = \frac{t^{n+1}}{n+1}\)[/tex], we get:
[tex]\[
\int 13t \, dt = 13 \left( \frac{t^2}{2} \right) = \frac{13}{2} t^2
\][/tex]
[tex]\[
\int 1.5t^2 \, dt = 1.5 \left( \frac{t^3}{3} \right) = 0.5 t^3
\][/tex]
Thus, the antiderivative of [tex]\( v(t) \)[/tex] is:
[tex]\[
\frac{13}{2} t^2 - 0.5 t^3
\][/tex]
5. Evaluate the definite integral:
We now evaluate the antiderivative from [tex]\( t = 0 \)[/tex] to [tex]\( t = 2 \)[/tex]:
[tex]\[
D = \left[ \frac{13}{2} t^2 - 0.5 t^3 \right]_{0}^{2}
\][/tex]
Substituting [tex]\( t = 2 \)[/tex]:
[tex]\[
\left( \frac{13}{2} (2)^2 - 0.5 (2)^3 \right) = \left( \frac{13}{2} \cdot 4 - 0.5 \cdot 8 \right) = \left( 26 - 4 \right) = 22
\][/tex]
Substituting [tex]\( t = 0 \)[/tex] (which just gives 0):
[tex]\[
\left( \frac{13}{2} (0)^2 - 0.5 (0)^3 \right) = 0
\][/tex]
6. Compute the total distance:
Therefore, the distance traveled by the car over the first 2 hours is:
[tex]\[
D = 22 - 0 = 22 \text{ miles}
\][/tex]
Thus, the car travels 22 miles during the first 2 hours.
