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Let's break down the problem step by step to find the correct equation.

1. **Total Time**: The student runs for a total of 45 minutes.
   - She runs at a constant rate of 8 miles per hour for **a** minutes.
   - Then, she slows to a constant rate of 7.5 miles per hour for **b** minutes.
   
   We know:
   \[
   a + b = 45
   \]
   This ensures that the total time spent running is 45 minutes.

2. **Distance Calculation**:
   - **Distance for the first part**: The student runs at 8 miles per hour for **a** minutes. To convert **a** minutes to hours, we divide by 60. So the distance covered during this time is:
   \[
   \text{Distance}_1 = 8 \times \frac{a}{60}
   \]
   
   - **Distance for the second part**: The student runs at 7.5 miles per hour for **b** minutes. Similarly, converting **b** minutes to hours, the distance covered in this part is:
   \[
   \text{Distance}_2 = 7.5 \times \frac{b}{60}
   \]

3. **Total Distance**: The total distance \(D\) is the sum of the distances from both parts:
   \[
   D = 8 \times \frac{a}{60} + 7.5 \times \frac{b}{60}
   \]

4. **Selecting the Correct Equation**:
   Now, let's match this with the given options:
   \[
   \boxed{8 \left( \frac{a}{60} \right) + 7.5 \left( \frac{b}{60} \right) = D}
   \]
   This matches the second option.

Thus, the correct answer is the **second option**.

Would you like further details or clarifications?

Here are five related questions for deeper understanding:
1. How would the equation change if the total running time was different?
2. What happens to the total distance if the student runs at the same speed throughout?
3. How can you convert minutes to hours when calculating speed in miles per hour?
4. How does the concept of rate relate to distance and time in these kinds of problems?
5. What if the student alternated between three different speeds? How would the equation change?

**Tip**: Always ensure units are consistent when applying formulas, especially when time is given in minutes but the rate is per hour.

A student on the track team runs 45 minutes each day as a part of her training. She begins her workout by running at a constant rate of 8 miles per hour for a minutes, then slows to a constant rate of 7.5 miles per hour for b minutes. Which equation describes the relationship between the distance she runs in miles, D, and her running speed, in miles per hour?

Solve a distance-time problem where a student runs for 45 minutes at two different speeds: 8 mph and 7.5 mph. Learn to formulate and solve equations based on distance-time relationships, ideal for students in Grades 8-10.

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