Let's solve the problem step-by-step to determine the function that describes the car's distance from Denver.
1. Initial Information:
- The car starts 300 miles away from Denver.
- After 3 hours, the car is 180 miles away from Denver.
2. Determine the Distance Covered:
- The car's initial distance from Denver is 300 miles.
- After 3 hours, the car is 180 miles from Denver.
- Therefore, the car has traveled [tex]\( 300 - 180 = 120 \)[/tex] miles in 3 hours.
3. Calculate the Speed:
- Speed = Distance / Time
- The distance covered by the car is 120 miles.
- The time taken is 3 hours.
- Hence, the speed of the car is [tex]\( \frac{120}{3} = 40 \)[/tex] miles per hour.
4. Establish the Relationship Between Distance and Time:
- We know the speed of the car, which is 40 miles per hour.
- The distance from Denver is decreasing as the car travels towards it, so the speed will have a negative sign in our equation.
- The initial distance from Denver is 300 miles.
5. Form the Equation:
- Let [tex]\( y \)[/tex] be the distance from Denver after [tex]\( x \)[/tex] hours.
- The relationship is linear, and can be expressed in the form of the equation:
[tex]\[
y = mx + b
\][/tex]
- Here, [tex]\( m \)[/tex] is the rate of change (speed), which is [tex]\(-40\)[/tex] miles per hour (negative because the distance is decreasing), and [tex]\( b \)[/tex] is the initial distance from Denver, which is 300 miles.
6. Write the Equation:
- Substituting [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the equation, we get:
[tex]\[
y = -40x + 300
\][/tex]
Therefore, the function that describes the car's distance from Denver is:
A. [tex]\(\mathbf{y = -40x + 300}\)[/tex]
So, the correct answer is A.
