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The question on the image is about Tyler biking for 30 minutes at different speeds. It asks to write equations that describe the relationship between Tyler's biking distance (in miles, \( D \)) and his speed (in miles per hour). Here's how we can break it down:

The problem asks us to consider various scenarios:

1. **Scenario a**: Tyler bikes at a constant speed of 10 miles per hour for the entire 30 minutes.
   
2. **Scenario b**: Tyler bikes at a constant speed of 12 miles per hour for the first 10 minutes and at 8 miles per hour for the last 20 minutes.

3. **Scenario c**: Tyler bikes at a constant speed of \( x \) miles per hour for the first 10 minutes and then at \( y \) miles per hour for the last 20 minutes.

The relationship between distance, speed, and time can be expressed using the formula:

\[
D = \text{Speed} \times \text{Time}
\]

### Solution to each scenario:

1. **Scenario a**: 
   - Time = 30 minutes = 0.5 hours
   - Speed = 10 miles per hour
   - \( D = 10 \times 0.5 = 5 \text{ miles} \)

2. **Scenario b**: 
   - First 10 minutes (0.167 hours) at 12 miles per hour:
     \[
     D_1 = 12 \times \frac{10}{60} = 12 \times 0.167 = 2 \text{ miles}
     \]
   - Last 20 minutes (0.333 hours) at 8 miles per hour:
     \[
     D_2 = 8 \times \frac{20}{60} = 8 \times 0.333 = 2.67 \text{ miles}
     \]
   - Total distance \( D = D_1 + D_2 = 2 + 2.67 = 4.67 \text{ miles} \)

3. **Scenario c**: 
   - First 10 minutes at \( x \) miles per hour:
     \[
     D_1 = x \times \frac{10}{60} = x \times 0.167 = 0.167x \text{ miles}
     \]
   - Last 20 minutes at \( y \) miles per hour:
     \[
     D_2 = y \times \frac{20}{60} = y \times 0.333 = 0.333y \text{ miles}
     \]
   - Total distance \( D = D_1 + D_2 = 0.167x + 0.333y \text{ miles} \)

These are the expressions that describe the distance \( D \) based on the time and speeds in each scenario.

Would you like further details on any of these steps or clarifications? Here are some related questions:

1. How do we derive the distance formula from speed and time?
2. What would happen if Tyler’s speed changes more than twice in 30 minutes?
3. How can we graph these relationships between speed and time?
4. What if the times given were in different units (e.g., seconds)?
5. Can you explain how to solve for \( x \) and \( y \) if a total distance is provided?

**Tip:** Always ensure that units are consistent when calculating time and distance (e.g., convert minutes to hours when using speeds in miles per hour).

Tyler bikes 30 minutes each day for exercise. Write an equation to describe the relationship between his distance in miles, D, and his biking speed, miles per hour, when he bikes: a) at a constant speed of 10 miles per hour for the entire 30 minutes, b) at a constant speed of 12 miles per hour for the first 10 minutes, then at 8 miles per hour for the last 20 minutes, c) at a constant speed of x miles per hour for the first 10 minutes, then at y miles per hour for the last 20 minutes.

Learn how to calculate the distance Tyler covers during his daily 30-minute biking routine. This problem explores scenarios with different biking speeds, applying the formula D = Speed × Time. Suitable for students in Grades 6-8.

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