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 scheduled
- \( S \) = number of spin classes scheduled

Given that each yoga class requires 5 participants and each spin class requires 10 participants, the total number of participants (\( P \)) can be represented by the inequality:

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

This inequality expresses the constraint that the total number of participants in the yoga and spin classes together must not exceed the maximum capacity of 200 participants.

2. **Determining if the Schedule Meets Capacity:**

The center schedules:
- \( Y = 6 \) (yoga classes)
- \( S = 8 \) (spin classes)

Substituting into the inequality:

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

Calculating the left side:

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

Now, we compare this to the maximum capacity:

\[
110 \leq 200
\]

Since 110 is indeed less than 200, **the schedule meets the participant's capacity.**