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Determine whether the relationship between the two quantities shown in each table or graph is linear. If so, find the constant rate of change. If not, explain your reasoning

To determine if the relationship between **Time (h)** and **Cost (¢)** in the table is linear, we can check if there is a constant rate of change between the two quantities.

### Table Data
| Time (h) | Cost (¢) |
|----------|----------|
| 5        | 15       |
| 8        | 24       |
| 12       | 36       |
| 24       | 72       |

### Steps to Check for Linearity
1. **Calculate the rate of change** between each consecutive pair of points.
   \[
   \text{Rate of Change} = \frac{\Delta \text{Cost}}{\Delta \text{Time}}
   \]

2. **Calculate for each interval**:
   - Between 5 hours and 8 hours:
     \[
     \frac{24 - 15}{8 - 5} = \frac{9}{3} = 3 \, \text{¢ per hour}
     \]
   - Between 8 hours and 12 hours:
     \[
     \frac{36 - 24}{12 - 8} = \frac{12}{4} = 3 \, \text{¢ per hour}
     \]
   - Between 12 hours and 24 hours:
     \[
     \frac{72 - 36}{24 - 12} = \frac{36}{12} = 3 \, \text{¢ per hour}
     \]

3. **Conclusion**: Since the rate of change (3¢ per hour) is constant for each interval, the relationship between **Time** and **Cost** is **linear**.

### Constant Rate of Change
The constant rate of change is **3¢ per hour**.

Would you like further details or have any questions?

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Here are some related questions that might help deepen your understanding:

1. How would the table change if the rate of change were doubled?
2. What would the total cost be for running the computer for 30 hours?
3. How can we represent this linear relationship as an equation?
4. What would happen to the rate of change if the costs were given in dollars instead of cents?
5. How can you determine if a relationship is linear from a graph instead of a table?

**Tip:** For any table of values, a constant rate of change between intervals is a quick indicator of linearity.

Determine whether the relationship between the two quantities shown in each table or graph is linear. If so, find the constant rate of change. If not, explain your reasoning.

Explore how to determine if the relationship between time and cost in a table is linear by calculating the constant rate of change. This example table shows the cost of running a computer over different time intervals, with a solution verifying linearity at 3¢ per hour. Suitable for middle school students studying linear relationships.

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