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 (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)
(/ points)

Answer 1

1. Variables Definition:

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

   Linear Inequality: The total number of participants can be expressed with the following inequality:

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

   Here, \( 5Y \) represents the number of participants in the yoga classes (5 participants per class), and \( 10S \) represents the number of participants in the spin classes (10 participants per class). The inequality states that the total number of participants from both class types must not exceed 200.

2. Checking the Schedule: The center schedules 6 yoga classes and 8 spin classes. Plugging these values into the inequality:

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

   Calculating each term:

   \[ 30 + 80 \leq 200 \]

   Simplifying:

   \[ 110 \leq 200 \]

   This statement is true. Therefore, the scheduled classes meet the participant's capacity, as the total number of participants (110) is less than the maximum allowed (200).