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)

(7 points)

Answer 1

### 1. Two-variable Linear Inequality

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

The total number of participants for yoga classes is given by \( 5Y \), since each yoga class requires 5 participants. The total number of participants for spin classes is \( 10S \), as each spin class requires 10 participants.

The fitness center can accommodate a maximum of 200 participants, which leads to the following inequality:

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

### 2. Determining if the Schedule Meets Capacity

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

We can substitute these values into the inequality:

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

Calculating this:

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

Now, add these two totals:

\[
30 + 80 = 110
\]

Now, we need to verify if this total meets the capacity restriction:

\[
110 \leq 200
\]

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