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

1. **Defining Variables and Creating the Inequality:**

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

Each yoga class requires 5 participants, so the total number of participants for yoga classes will be \( 5Y \). Each spin class requires 10 participants, which means that the total number of participants for spin classes will be \( 10S \). Since the fitness center can accommodate a maximum of 200 participants, the linear inequality representing this constraint can be written as:

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

2. **Evaluating the Scheduled Classes:**

The center is scheduling 6 yoga classes and 8 spin classes. We can substitute \( Y = 6 \) and \( S = 8 \) into the inequality:

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

Calculating this gives:

\[ 30 + 80 = 110 \]

Now we need to compare the total participants (110) to the maximum capacity (200):

Since \( 110 \leq 200 \), the schedule of 6 yoga classes and 8 spin classes meets the participant capacity requirement.