Question

Rain fell at a steady rate of 2 inches every 3 hours.
Complete the table to describe the relationship.


Time(h)
3




12
Rainfall (in.)


4
6



Graph the data in the table on the coordinate plane grid. Draw the line.

Find slope
State the meaning of the slope in context.

Answer 1

time(h) 3 _ _ 12
Rainfall (in.) _ 4 6 _

Answer 2

To complete the table and analyze the relationship between time and rainfall, we start with the provided steady rate of rainfall, which is 2 inches every 3 hours.

### Step 1: Complete the Table

**Given:**
- Rainfall rate = 2 inches per 3 hours

**Calculating rainfall for the specified times:**

1. **For 3 hours:**

\[
\text{Rainfall} = \frac{2 \text{ inches}}{3 \text{ hours}} \times 3 \text{ hours} = 2 \text{ inches}
\]

2. **For 6 hours:**

\[
\text{Rainfall} = \frac{2 \text{ inches}}{3 \text{ hours}} \times 6 \text{ hours} = 4 \text{ inches}
\]

3. **For 12 hours:**

\[
\text{Rainfall} = \frac{2 \text{ inches}}{3 \text{ hours}} \times 12 \text{ hours} = 8 \text{ inches}
\]

### Completed Table

| Time (hours) | Rainfall (inches) |
|--------------|--------------------|
| 3 | 2 |
| 6 | 4 |
| 12 | 8 |

### Step 2: Calculate Slope

The slope \( m \) of a line on a coordinate grid can be found using the formula:

\[
m = \frac{\Delta y}{\Delta x}
\]

Where \( \Delta y \) is the change in the y-values (rainfall) and \( \Delta x \) is the change in the x-values (time).

Using the points (3, 2) and (12, 8):

\[
\Delta y = 8 - 2 = 6 \quad \text{(change in rainfall)}
\]
\[
\Delta x = 12 - 3 = 9 \quad \text{(change in time)}
\]

Now plug in the values into the slope formula:

\[
m = \frac{6}{9} = \frac{2}{3}
\]

### Step 3: Meaning of the Slope in Context

The slope of \( \frac{2}{3} \) means that for every hour (1 hour) that passes, an additional \( \frac{2}{3} \) of an inch of rainfall occurs. In other words, it represents the constant rate of rainfall over time: it indicates that the rain is falling at a steady rate of approximately 0.67 inches per hour.

### Step 4: Graph the Data Points

To create the graph:

1. Mark the x-axis (Time in hours) and y-axis (Rainfall in inches).
2. Plot the points:
- (3, 2)
- (6, 4)
- (12, 8)
3. Draw a straight line through the points.

### Graph Representation

While I'm unable to directly create visual graphs, here's how you would visualize it:

- Draw a coordinate plane with Time on the horizontal axis and Rainfall on the vertical axis.
- Place dots at each of the points you calculated.
- Connect the dots with a straight line to show the relationship.

This would effectively represent the data and the steady increase in rainfall over time based on the given rate.

Answer 3

Let's complete the table with the given information. We need to fill in the missing values for time and rainfall based on the steady rate of rainfall, which is 2 inches every 3 hours.

### Given Information
- **Rate of rainfall**: 2 inches every 3 hours

### Step 1: Complete the Table

We have the known values:

- **For 3 hours**: We will calculate rainfall.
- **At 6 inches**: We need to find the corresponding time.
- **At 4 inches**: We need to find the corresponding time.
- **For 12 hours**: We will calculate rainfall.

#### Calculating Missing Values

1. **For 3 hours**:
\[
\text{Rainfall} = \frac{2 \text{ inches}}{3 \text{ hours}} \times 3 \text{ hours} = 2 \text{ inches}
\]

2. **Finding Time for 4 inches**:
\[
\text{If } y \text{ (rainfall) is 4 inches:}
\]
- The formula relating rainfall to time is the same:
\[
\text{Amount of rainfall} = \frac{2 \text{ inches}}{3 \text{ hours}} \times t \text{ (time in hours)}
\]
Setting rainfall to 4 inches:
\[
4 = \frac{2}{3} t
\]
Multiplying both sides by 3:
\[
12 = 2t \quad \Rightarrow \quad t = \frac{12}{2} = 6 \text{ hours}
\]

3. **Finding Time for 6 inches**:
\[
\text{If } y \text{ (rainfall) is 6 inches:}
\]
\[
6 = \frac{2}{3} t
\]
Multiplying both sides by 3:
\[
18 = 2t \quad \Rightarrow \quad t = \frac{18}{2} = 9 \text{ hours}
\]

4. **For 12 hours**:
\[
\text{Rainfall} = \frac{2 \text{ inches}}{3 \text{ hours}} \times 12 \text{ hours} = 8 \text{ inches}
\]

### Completed Table

Now we can fill out the missing values in the table:

| Time (h) | Rainfall (in.) |
|----------|-----------------|
| 3 | 2 |
| 6 | 4 |
| 9 | 6 |
| 12 | 8 |

### Step 2: Graph the Data

To graph the data:

1. Draw a coordinate plane with “Time (h)” on the x-axis and “Rainfall (in.)” on the y-axis.
2. Plot the points:
- (3, 2)
- (6, 4)
- (9, 6)
- (12, 8)
3. Draw a straight line through the points to indicate the relationship.

### Step 3: Calculate the Slope

Using the points (3, 2) and (12, 8):

\[
\Delta y = 8 - 2 = 6
\]
\[
\Delta x = 12 - 3 = 9
\]
\[
m = \frac{\Delta y}{\Delta x} = \frac{6}{9} = \frac{2}{3}
\]

### Interpretation of the Slope

The slope of \( \frac{2}{3} \) indicates that for every hour that passes, approximately \( \frac{2}{3} \) inches of rain falls. This represents the constant rate of rainfall over time. Thus, the interpretation is that the rain is consistently falling at a rate of about 0.67 inches per hour.