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The table shows the cost associated with different amounts of minutes, $ m $, along with two questions:

### Table:
| Minutes, $ m $ | Cost, $ \$ $ |
|------------------|----------------|
| 500              | 26             |
| 1,000            | 38             |
| 1,500            | 50             |
| 2,000            | 62             |
| 2,500            | 74             |
| 3,000            | 86             |

### Questions:
- (a) From $ m = 500 $ to $ m = 2,000 $
- (b) From $ m = 1,000 $ to $ m = 3,000 $

It seems like the question is asking about the rate of change (possibly average rate of change) or the total cost difference over these ranges.

Let's analyze each part:

### (a) From $ m = 500 $ to $ m = 2,000 $:
- At $ m = 500 $, the cost is $ 26 $.
- At $ m = 2,000 $, the cost is $ 62 $.

To calculate the average rate of change (or the slope) between these two points:
\[
\text{Average rate of change} = \frac{\Delta \text{Cost}}{\Delta m} = \frac{62 - 26}{2,000 - 500} = \frac{36}{1,500} = 0.024
\]
The average rate of change is \$0.024 per minute.

### (b) From $ m = 1,000 $ to $ m = 3,000 $:
- At $ m = 1,000 $, the cost is $ 38 $.
- At $ m = 3,000 $, the cost is $ 86 $.

The average rate of change here is:
\[
\text{Average rate of change} = \frac{86 - 38}{3,000 - 1,000} = \frac{48}{2,000} = 0.024
\]
Again, the average rate of change is \$0.024 per minute.

Both parts (a) and (b) give the same average rate of change, which suggests a constant rate of increase in cost over time.

Would you like more details or have any questions?

### Related Questions:
1. How do you calculate the total cost increase over a given range of minutes?
2. How can the rate of change help predict future costs?
3. Can you find the cost at $ m = 4,000 $ using the rate of change?
4. Is this linear relationship applicable for values of $ m $ beyond 3,000 minutes?
5. What would the total cost be if the cost per minute was $ \$0.03 $ instead of $ \$0.024 $?

### Tip:
The average rate of change is a helpful tool for finding trends in data over an interval, especially for linear relationships like this one!

The table shows the cost associated with different amounts of minutes, m, along with two questions:
- (a) from m = 500 to m = 2,000
- (b) from m = 1,000 to m = 3,000

Explore how to calculate the average rate of change in cost from m = 500 to m = 2,000 and m = 1,000 to m = 3,000. Learn to apply the formula for rate of change to real-world linear relationships with step-by-step solutions. Suitable for Grades 8-10.

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