Let's analyze the problem step by step.

Part (a): Determine the slope of the line

The table provides two points:

• Point 1: $(0, 65)$, where the car starts braking at 65 mph.
• Point 2: $(130, 0)$, where the car comes to a stop after traveling 130 feet.

To calculate the slope ($m$): $m = \frac{\text{change in speed}}{\text{change in distance}} = \frac{0 - 65}{130 - 0} = \frac{-65}{130} = -0.5$

The slope is $-0.5$, meaning the car's speed decreases by 0.5 mph for every 1 foot it travels after braking.

Part (b): Determine the equation of the line

The general equation of a line is: $v = m \cdot d + b$ where:

• $m = -0.5$ (slope from part a),
• $b = 65$ (the initial speed when $d = 0$).

Substituting these values: $v = -0.5 \cdot d + 65$

This is the equation for the speed of the car as a function of the distance traveled after braking.

Part (c): Determine the speed at $d = 60$

Using the equation from part (b): $v = -0.5 \cdot d + 65$ Substitute $d = 60$: $v = -0.5 \cdot 60 + 65 = -30 + 65 = 35$

The car is traveling at 35 mph when it is 60 feet from where the brakes were applied.

Final Answers:

• (a): The slope is $-0.5$, meaning the speed decreases by 0.5 mph for each foot traveled.
• (b): The equation is $v = -0.5d + 65$.
• (c): The speed at $d = 60$ is 35 mph.

Would you like a detailed graph or further clarification? Here are some related questions to explore:

1. What is the significance of a negative slope in real-world scenarios like this?
2. How would the graph of this function look in terms of speed vs. distance?
3. If the car continued at this rate, at what distance would its speed become negative?
4. How does the concept of linear functions apply in physics problems like this one?
5. What are potential sources of error when modeling braking with a linear function?

Tip: When working with linear equations, always double-check the slope's sign to confirm the direction of change in real-world contexts.