https://file.aiquickdraw.com/mathbot/file/1730450065229stodd85m.pdf

I've extracted the text from your PDF, which contains several tasks related to planning a road network based on linear functions and height differences. Here’s a summary of the tasks:

1. **Determine the linear function** for each road segment connecting a roundabout at point (75, 25) to three cities (A, B, C). Include the slope and y-intercept for each function.
   
2. **Write the domain and range** for the three functions.
   
3. **Draw the road network** in a coordinate system, showing the roundabout and roads to cities A, B, and C.
   
4. **Identify which road segment is the steepest.**
   
5. **Calculate the slope** for a new road to city D at point (25075, 105) and discuss its impact on the overall network.
   
6. **Calculate the lengths** of the four road segments.

### Let's tackle these tasks step-by-step.

#### Task 1: Determine the Linear Functions

To find the linear function between the roundabout and each city, we can use the formula for the slope \( m \):

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

And the linear equation in slope-intercept form:

\[
y = mx + b
\]

where \( b \) is the y-intercept.

- **City A**: (5075, 75)
- **City B**: (14925, -5)
- **City C**: (35075, 145)

Let's calculate these functions now.Here are the linear functions for each road segment connecting the roundabout to the cities:

1. **Road to City A** (5075, 75):
   - **Slope (m)**: 0.01
   - **Y-intercept (b)**: 24.25
   - **Function**: \( U = 0.01T + 24.25 \)

2. **Road to City B** (14925, -5):
   - **Slope (m)**: -0.00202
   - **Y-intercept (b)**: 25.15
   - **Function**: \( U = -0.00202T + 25.15 \)

3. **Road to City C** (35075, 145):
   - **Slope (m)**: 0.00343
   - **Y-intercept (b)**: 24.74
   - **Function**: \( U = 0.00343T + 24.74 \)

### Task 2: Domain and Range

For linear functions like these, the domain will typically be all real numbers, but since we're dealing with physical distances, we can restrict it to the range of distances between the roundabout and each city. Let's determine the ranges for each function based on the given coordinates.

- **City A**: 0 to 5000 meters (0 to 5 km)
- **City B**: 0 to 14850 meters (0 to -15 km)
- **City C**: 0 to 35000 meters (0 to 35 km)

Let's formalize this for each function.Here are the domains and ranges for the linear functions connecting the roundabout to each city:

1. **City A**:
   - **Domain**: \( T \in [0, 5000] \)
   - **Range**: \( U \in [24.25, 75] \)

2. **City B**:
   - **Domain**: \( T \in [0, 14850] \)
   - **Range**: \( U \in [25.15, -5] \)

3. **City C**:
   - **Domain**: \( T \in [0, 35000] \)
   - **Range**: \( U \in [24.74, 145] \)

### Task 3: Draw the Road Network

To visualize the road network, I'll plot the roundabout and the roads to cities A, B, and C. Let's create that plot.It seems like I can’t do more advanced data analysis right now. Please try again later. If you have any specific questions or need help with another aspect of the tasks, let me know!

Explore how to plan a road network connecting a roundabout to three cities using linear functions. This task includes determining slopes, writing domain and range, and calculating road segment lengths, making it suitable for high school students.

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