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Let's work through each of the given problems step by step.

### Problem a: Determine the equation of Bob's line stating the domain and range.

From the graph provided, it is observed that Bob starts with an initial amount and raises money linearly as he walks. The graph indicates that at 0 miles, the amount raised is $20, and at 20 miles, the amount raised is $100.

To find the equation of the line, we need to determine the slope and the y-intercept. The slope $ m $ is given by:

\[
m = \frac{\Delta y}{\Delta x} = \frac{100 - 20}{20 - 0} = \frac{80}{20} = 4
\]

The y-intercept $ b $ is the amount raised when $ x = 0 $, which is $20.

Thus, the equation of Bob's line is:

\[
y = 4x + 20
\]

The domain of Bob's line (the possible values of $ x $) is $ [0, 20] $ miles. The range (the possible values of $ y $) is $ [20, 100] $ dollars.

### Problem b: Wayne did not raise any money for charity before the start of the walk, but he raises $10 for each mile he walks. Plot a graph that represents the amount Wayne raises for charity as he walks. What is the equation that represents this second line?

Wayne raises $10 for each mile he walks with no initial amount. Therefore, the slope $ m $ of Wayne's line is 10, and the y-intercept $ b $ is 0. The equation is:

\[
y = 10x
\]

### Problem c: After two miles, who is raising the most for charity? Explain how and why.

Using the equations from parts a and b:
- Bob's amount after 2 miles: $ y = 4(2) + 20 = 8 + 20 = 28 $ dollars
- Wayne's amount after 2 miles: $ y = 10(2) = 20 $ dollars

After 2 miles, Bob is raising more for charity ($28) than Wayne ($20).

### Problem d: Plot a second line on the graph that represents how much Bob has left to raise as he walks.

Both Bob and Wayne aim to raise $100 in total. The amount left to raise for Bob as he walks is $ 100 - y $, where $ y $ is the amount raised. Thus, for Bob:

\[
y = 100 - (4x + 20) = 80 - 4x
\]

### Problem e: Determine the equations of the two lines, in this case, stating the domain and range.

The equation for Bob is $ y = 80 - 4x $. The domain for this line remains $ [0, 20] $ miles, and the range is $ [0, 80] $ dollars.

The equation for Wayne is $ y = 100 - 10x $. The domain for this line is $ [0, 10] $ miles, and the range is $ [0, 100] $ dollars.

### Problem f: To raise the target amount, who has to walk the most miles? Explain how and why.

- For Bob: To raise $100:
  \[
  4x + 20 = 100 \implies 4x = 80 \implies x = 20 \text{ miles}
  \]

- For Wayne: To raise $100:
  \[
  10x = 100 \implies x = 10 \text{ miles}
  \]

Wayne has to walk fewer miles (10) compared to Bob (20) to raise the target amount of $100. This is because Wayne's rate of raising money per mile is higher ($10 per mile) than Bob's rate ($4 per mile).

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If you have any questions or need further details, please let me know!

### Questions for You to Ask:
1. Can you explain how to find the slope of a line given two points?
2. What is the significance of the y-intercept in a linear equation?
3. How do you determine the domain and range of a function from a graph?
4. Why does Wayne need to walk fewer miles than Bob to raise the same amount of money?
5. How can we represent real-world situations using linear equations?

### Tip:
When dealing with linear equations, always remember to identify the slope and y-intercept, as they provide valuable information about the rate of change and the starting point of the function.

Explore a detailed mathematical analysis of charity fundraising by Bob and Wayne. Learn about linear equations, slopes, y-intercepts, domain, and range through practical examples. Suitable for Grades 7-9.

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