To determine the centripetal acceleration of the race car, we'll use the concept of centripetal force and acceleration in circular motion. Centripetal acceleration is given by the formula:
[tex]\[ a = \frac{v^2}{r} \][/tex]
where:
- [tex]\( v \)[/tex] is the velocity of the object.
- [tex]\( r \)[/tex] is the radius of the circular path.
Given:
- The velocity ([tex]\( v \)[/tex]) of the race car is [tex]\( 135 \)[/tex] miles per hour.
- The radius ([tex]\( r \)[/tex]) of the track is [tex]\( 0.450 \)[/tex] miles.
Now, we'll substitute the given values into the formula to find the centripetal acceleration.
[tex]\[ a = \frac{(135 \, \text{mi/hr})^2}{0.450 \, \text{mi}} \][/tex]
First, we square the velocity:
[tex]\[ (135 \, \text{mi/hr})^2 = 135 \times 135 = 18225 \, \text{(mi/hr)}^2 \][/tex]
Next, we divide this by the radius:
[tex]\[ a = \frac{18225 \, \text{(mi/hr)}^2}{0.450 \, \text{mi}} \][/tex]
Perform the division:
[tex]\[ a = 40500 \, \text{mi/hr}^2 \][/tex]
Thus, the centripetal acceleration of the car is:
[tex]\[ \boxed{40500 \text{ mi/hr}^2} \][/tex]
Comparing with the given options, the correct centripetal acceleration is:
[tex]\[ 40,500 \, \text{mi/hr}^2 \][/tex]
So, the answer is:
[tex]\[ \boxed{40500 \, \text{mi/hr}^2} \][/tex]
