Let's break down and solve each of the problems on the sheet:

1. Benjamin Hiking:

You are asked to find the slope of the graph for Benjamin's hike and the unit rate.

• The graph shows distance (miles) on the y-axis and time (hours) on the x-axis. To find the slope:

  $\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}$

  From the graph, a point appears to be (0, 0) and another around (10, 10), so:

  $\text{slope} = \frac{10 - 0}{10 - 0} = 1$

• The unit rate is the slope: 1 mile per hour.

2. Marcy Hiking:

For Marcy, the table provides values for time (hours) and distance (miles). To find the slope:

We can pick two points from the table: (5, 6) and (10, 12). The slope is:

$\text{slope} = \frac{12 - 6}{10 - 5} = \frac{6}{5} = 1.2$

So the slope is 1.2 miles per hour, which is also the unit rate.

3. Laura vs. Piyush Walking:

• Laura: The equation $y = 3.5x$ tells us that Laura's walking rate is 3.5 miles per hour (the coefficient of $x$).

• Piyush: The graph shows distance on the y-axis and time on the x-axis. Using two points from the graph, say (0, 0) and (10, 20):

  $\text{slope} = \frac{20 - 0}{10 - 0} = 2 \quad \text{miles per hour}$

  Therefore, Laura walks faster than Piyush (3.5 mph vs. 2 mph).

4. Rainfall:

a. The problem states rain falls at 2 inches every 3 hours, so we can complete the table:

• For 6 hours: $\frac{6}{3} \times 2 = 4$ inches
• For 12 hours: $\frac{12}{3} \times 2 = 8$ inches

The completed table is:

b. For the graph, plot the points (3, 2), (6, 4), and (12, 8) on the provided grid, then draw a line through them.

c. To find the slope of the rainfall:

$\text{slope} = \frac{2 - 0}{3 - 0} = \frac{2}{3}$

d. The unit rate is the slope: $\frac{2}{3}$ inches per hour.

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

1. How would the graph for Marcy look if her speed doubled?
2. What happens to the slope if Benjamin hikes at a constant speed of 2 miles per hour instead?
3. Can you explain how to identify the slope from a table instead of a graph?
4. If the rainfall increased to 3 inches every 3 hours, how would the slope change?
5. How would you interpret the unit rate in real-life contexts for these problems?

Tip: Always ensure that both your "rise" (change in y) and "run" (change in x) are measured between clear points when calculating slope.