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)

1 answer

1. Defining Variables and Creating a Linear Inequality

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

Inequality: Each yoga class requires 5 participants, so the total number of participants from yoga classes is \( 5Y \). Each spin class requires 10 participants, contributing \( 10S \) to the total. The total number of participants must be less than or equal to the fitness center's maximum capacity of 200 participants. Therefore, we can express this as a linear inequality:

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

2. Checking the Scheduled Classes Against the Capacity

The center schedules \( Y = 6 \) yoga classes and \( S = 8 \) spin classes. We can substitute these values into the inequality from part 1:

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

Calculating the left side:

\[ 5(6) = 30 \] \[ 10(8) = 80 \] Adding these together:

\[ 30 + 80 = 110 \]

Now, we compare this total to the maximum capacity:

\[ 110 \leq 200 \]

Since 110 is less than 200, the scheduled classes do meet the participant's capacity.