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

To solve this problem, we need to define the variables and create the appropriate linear inequality based on the given information.

1. Define the Variables and Create the Inequality

Let:

• \( Y \) = number of yoga classes scheduled
• \( S \) = number of spin classes scheduled

Next, we need to consider the number of participants each class requires:

• Each yoga class requires 5 participants.
• Each spin class requires 10 participants.

The total number of participants for yoga and spin classes can be modeled with the inequality:

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

This inequality states that the total number of participants from both yoga and spin classes cannot exceed the maximum capacity of the fitness center, which is 200 participants.

2. Evaluate the Schedule Against the Inequality

Now, we need to check if the schedule of 6 yoga classes and 8 spin classes meets the participant capacity.

Substituting \( Y = 6 \) and \( S = 8 \) into the inequality:

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

Calculating the left side:

\[ 30 + 80 = 110 \]

Thus, we get:

\[ 110 \leq 200 \]

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

Conclusion: The scheduled classes are within the maximum participant limit of the fitness center.