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

Question

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

Let's define the variables first:

• Let \( Y \) represent the number of yoga classes scheduled.
• Let \( S \) represent the number of spin classes scheduled.

Step 1: Create the Two-Variable Linear Inequality

Each yoga class requires 5 participants, which means the total number of participants from yoga classes is \( 5Y \).

Each spin class requires 10 participants, leading to a total of \( 10S \) participants from spin classes.

The maximum number of participants that the center can accommodate is 200. Thus, the inequality representing the total number of participants is:

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

Step 2: Evaluate the Scheduled Classes

Now, let's determine if the currently scheduled classes meet the participant's capacity.

With:

• \( Y = 6 \) (yoga classes)
• \( S = 8 \) (spin classes)

Substituting these values into the inequality, we calculate:

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

Calculating:

• \( 5(6) = 30 \) (from yoga classes)
• \( 10(8) = 80 \) (from spin classes)

Now, add these together:

\[ 30 + 80 = 110 \]

Compare with Maximum Capacity

Now we compare the total participants with the maximum capacity:

\[ 110 \leq 200 \]

Since this statement is true, the schedule of 6 yoga classes and 8 spin classes meets the participant's capacity of 200.