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

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.