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.
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.

Answer 1

To tackle this problem, we will follow the instructions step-by-step.

### Step 1: Define Variables and Create a Linear Inequality

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

We know that:
- Each yoga class requires 5 participants.
- Each spin class requires 10 participants.
- The maximum capacity for participants is 200.

The total number of participants from yoga and spin classes can be represented by the inequality:

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

### Step 2: Determine if the Scheduled Classes Meet Capacity

Now, the center schedules:
- \( Y = 6 \) (yoga classes)
- \( S = 8 \) (spin classes)

We can now substitute these values into the inequality we created:

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

Calculating the left-hand side:

\[
5(6) = 30 \\
10(8) = 80 \\
30 + 80 = 110
\]

Now we can check the inequality:

\[
110 \leq 200
\]

Since 110 is less than 200, this schedule does indeed meet the participant's capacity.

### Conclusion
1. The two-variable linear inequality representing the constraint is \( 5Y + 10S \leq 200 \).
2. The scheduled classes (6 yoga classes and 8 spin classes) accommodate 110 participants, which is within the maximum capacity of 200 participants. Thus, the schedule meets the capacity requirement.