Let's analyze the uploaded image and solve the problem step-by-step.

Analysis of the Problem

Given Graph

• The graph shows the relationship between rain depth (inches) on the y-axis and time (hours) on the x-axis.
• The line is straight and goes through points (0, 0) and (5, 2.5), indicating a linear relationship.

Solutions

(a) In words, describe the function shown on the graph.

The function shows a steady increase in rain depth over time. As time progresses, the amount of rain increases at a constant rate. This linear relationship implies that rain is falling at a uniform rate.

(b) Find the slope of the graph and express it as a rate of change.

The slope of a line is calculated using the formula:
$\text{slope} = \frac{\Delta y}{\Delta x} = \frac{\text{Change in Rain Depth}}{\text{Change in Time}}$

Using the points $(0, 0)$ and $(5, 2.5)$: $\text{slope} = \frac{2.5 - 0}{5 - 0} = \frac{2.5}{5} = 0.5$

Thus, the rate of change is 0.5 inches per hour.

(c) Briefly discuss the conditions under which a linear function is a realistic model for the given situation.

A linear function is a realistic model in situations where the rate of rainfall remains constant over time. This means:

• The rain falls uniformly, without sudden increases or decreases in intensity.
• External factors (e.g., changes in weather patterns) do not affect the rainfall rate.
• The relationship between time and rain depth continues to follow a straight line.

Final Answer Selection

From the options provided, the correct answer is:
A. It is a good model if the rate of rainfall increases over time.

Would you like further explanation or have any questions?
Here are some follow-up questions to deepen understanding:

1. What happens to the graph if the rainfall rate changes over time?
2. How can you tell if a situation is linear or nonlinear by looking at a graph?
3. What real-life conditions might make this linear model invalid?
4. How would the slope change if the rain fell faster?
5. What units are associated with the slope in this context?

Tip: Always check the units when calculating slope to ensure proper interpretation.