Let's go through each part of this question step by step based on the provided image.

Part (a)

The question asks about the change in input and the corresponding change in output required to move to an input of 3.5 hours. The rate of change (ROC) appears to be defined as 5 miles per hour (5 mph).

To determine the required changes:

1. Change in Input Calculation: You need to go from the current input (possibly 2 hours based on the graph) to 3.5 hours. The change in input, ΔI, would be: $\Delta I = 3.5 - 2 = 1.5 \text{ hours}$

2. Change in Output Calculation: Since the rate of change is 5 mph, the change in output, ΔO, would be calculated using: $\Delta O = \text{Rate of Change} \times \Delta I = 5 \times 1.5 = 7.5 \text{ miles}$

So, to reach an input of 3.5 hours, the output (miles) should increase by 7.5 miles.

Part (b)

Using the changes found in part (a) to calculate the rate of change:

The rate of change (ROC) is defined as the change in output divided by the change in input: $\text{ROC} = \frac{\Delta O}{\Delta I} = \frac{7.5}{1.5} = 5 \text{ mph}$

This aligns with the given rate of change of 5 mph.

Part (c)

The answer in part (b) (5 mph) should match the given ROC stated in part (a). This consistency confirms that the calculated rate of change aligns with the problem's parameters.

Part (d)

The question here involves calculating the change in input and output that would produce a point 3 hours to the right of the given point (2, 25).

1. Change in Input: Moving 3 hours to the right means: $\Delta I = 3$

2. Change in Output: Using the rate of change, 5 mph: $\Delta O = 5 \times 3 = 15 \text{ miles}$

The new point would be: $(2 + 3, 25 + 15) = (5, 40)$

Directional arrows on the graph should show movement from (2, 25) to (5, 40).

Would you like further clarification on any part, or is there a specific part you'd like a deeper explanation for?

Here are some additional questions that extend this problem:

1. If the rate of change increased to 6 mph, how would this affect the answers in parts (a) and (d)?
2. What is the significance of a constant rate of change in the context of a linear graph?
3. How would you calculate the total distance traveled if the driving rate changes every hour?
4. What other real-world scenarios could this type of rate of change calculation apply to?
5. If the starting point was different, say (1, 10), how would it impact the calculations in parts (a) and (d)?

Tip: When working with rate of change problems, always ensure that the units in both input and output changes align to maintain accuracy in your calculations.