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.
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.
Answers
GPT-4o mini
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.
### 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.
time(h) 3 _ _ 12
Rainfall (in.) _ 4 6 _
Rainfall (in.) _ 4 6 _
GPT-4o mini
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.
### 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.