The question is asking to find the average speed for the entire journey described in the image. Here's how to solve it step by step:

Formula for Average Speed:

$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$

Steps:

1. Calculate Total Distance:

• Mr. Khadem drives:
  • 28 miles to work.
  • 14 miles to the grocery store.
  • 32 miles back home.
• Total distance: $\text{Total Distance} = 28 + 14 + 32 = 74 \, \text{miles}.$

2. Calculate Total Time:

The time for each leg of the journey is given by: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

• First leg (to work): $\text{Time}_1 = \frac{28}{59} \, \text{hours}.$

• Second leg (to grocery store): $\text{Time}_2 = \frac{14}{42} \, \text{hours}.$

• Third leg (back home): $\text{Time}_3 = \frac{32}{46} \, \text{hours}.$

• Total time: $\text{Total Time} = \text{Time}_1 + \text{Time}_2 + \text{Time}_3.$

3. Plug Values and Compute:

1. Compute each time: $\text{Time}_1 = \frac{28}{59} \approx 0.4746 \, \text{hours}.$ $\text{Time}_2 = \frac{14}{42} = 0.3333 \, \text{hours}.$ $\text{Time}_3 = \frac{32}{46} \approx 0.6957 \, \text{hours}.$

2. Add the times: $\text{Total Time} = 0.4746 + 0.3333 + 0.6957 = 1.5036 \, \text{hours}.$

3. Average Speed: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{74}{1.5036} \approx 49.2 \, \text{miles per hour}.$

Final Answer:

The average speed for the whole journey is 49.2 miles per hour.

Do you have any questions or want a detailed breakdown of any step? Here are 5 related questions to deepen understanding:

1. What is the difference between average speed and average velocity?
2. How do changes in speed affect the total time in a journey?
3. Can you derive the formula for average speed for a multi-leg journey?
4. What happens to the average speed if one leg of the journey is much slower than the others?
5. How would the calculations change if one of the speeds were in kilometers per hour?

Tip: Always pay attention to whether you’re asked for average speed or average velocity. Velocity considers direction, while speed does not.