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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.

3 answers

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:

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

  1. For 6 hours:

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

  1. 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 _

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.