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To determine the monthly fee for Colton's gym membership, we can analyze the relationship between $ t $ (the number of months) and $ C $ (the total cost) by calculating the rate of change from the values in the table. Given that the relationship is linear, the rate of change (or slope) will represent the monthly fee.

### Step 1: Calculate the Monthly Fee (Rate of Change)
The formula for the rate of change (slope) between two points $(t_1, C_1)$ and $(t_2, C_2)$ is:
\[
\text{slope} = \frac{C_2 - C_1}{t_2 - t_1}
\]

Let's use two pairs of points to calculate the monthly fee:
1. Using points $(t_1, C_1) = (2, 165)$ and $(t_2, C_2) = (5, 225)$:
   \[
   \text{slope} = \frac{225 - 165}{5 - 2} = \frac{60}{3} = 20
   \]

This result tells us that the monthly fee is $20$ dollars per month.

### Step 2: Confirm with Another Pair of Points
To ensure accuracy, let's verify this with another pair of points:
- Using $(t_1, C_1) = (5, 225)$ and $(t_2, C_2) = (7, 265)$:
  \[
  \text{slope} = \frac{265 - 225}{7 - 5} = \frac{40}{2} = 20
  \]

Both calculations yield a monthly fee of $20$ dollars, confirming that this is the correct rate.

### Conclusion
The monthly fee to remain a member is **$20 per month**.

Would you like a deeper explanation, or do you have any questions?

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Here are some related questions for further exploration:
1. How can we determine the initial one-time fee to join the gym?
2. How would the total cost formula for $C$ change if the monthly fee was variable instead of fixed?
3. What would be the total cost for a 12-month membership?
4. How would the answer change if there was an additional monthly surcharge?
5. Can you use the slope-intercept form of a line to model this situation?

**Tip:** The slope (rate of change) between two points in a linear relationship can often reveal the unit rate or recurring cost in real-life scenarios.

Colton is signing up for a gym membership with a one-time fee to join and then a monthly fee to remain a member. Let C represent the total cost of the gym membership over t months. The table below has select values showing the linear relationship between t and C. Determine the monthly fee to remain a member.

Learn how to determine the monthly fee for a gym membership with a linear relationship between time and cost. Using values provided in a table, we calculate the rate of change to find the recurring monthly cost. Suitable for students in Grades 8-10.

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