Question

Question A fitness center is hosting a special promotion for group classes. The center can accommodate a maximum of 200 participants. The center is offering two types of classes: yoga (Y) and spin (S). Each yoga class requires 5 participants to run, while each spin class requires 10 participants 1. Create a two-variable linear inequality that represents the constraint of total number of participants. Clearly define your variables (5 points) 2. The center schedules 6 yoga classes and 8 spin classes. Using the inequality created in part 1 determine if this schedule meets the participant's capacity. (2 points)

Answer 1

To solve this problem, we need to define the variables and create the appropriate linear inequality based on the given information.

### 1. Define the Variables and Create the Inequality

Let:
- \( Y \) = number of yoga classes scheduled
- \( S \) = number of spin classes scheduled

Next, we need to consider the number of participants each class requires:
- Each yoga class requires 5 participants.
- Each spin class requires 10 participants.

The total number of participants for yoga and spin classes can be modeled with the inequality:

\[
5Y + 10S \leq 200
\]

This inequality states that the total number of participants from both yoga and spin classes cannot exceed the maximum capacity of the fitness center, which is 200 participants.

### 2. Evaluate the Schedule Against the Inequality

Now, we need to check if the schedule of 6 yoga classes and 8 spin classes meets the participant capacity.

Substituting \( Y = 6 \) and \( S = 8 \) into the inequality:

\[
5(6) + 10(8) \leq 200
\]

Calculating the left side:

\[
30 + 80 = 110
\]

Thus, we get:

\[
110 \leq 200
\]

Since 110 is indeed less than 200, the schedule of 6 yoga classes and 8 spin classes meets the participant capacity.

**Conclusion**: The scheduled classes are within the maximum participant limit of the fitness center.