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.
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.
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.
Answers
GPT-4o mini
Let's define the variables and then create the two-variable linear inequality based on the provided information.
1. **Defining Variables:**
- Let \( Y \) represent the number of participants in yoga classes.
- Let \( S \) represent the number of participants in spin classes.
**Creating the Inequality:**
Given the constraints:
- Each yoga class requires 5 participants, so if there are \( y \) yoga classes, the total number of participants in yoga would be \( 5y \).
- Each spin class requires 10 participants, so if there are \( s \) spin classes, the total number of participants in spin would be \( 10s \).
The total number of participants for both yoga and spin classes can be represented as:
\[
5y + 10s \leq 200
\]
2. **Determining if the Scheduled Classes Meet Capacity:**
The center schedules 6 yoga classes and 8 spin classes. We can substitute these values into the inequality created in part 1.
- Number of yoga participants: \( 5y = 5 \times 6 = 30 \)
- Number of spin participants: \( 10s = 10 \times 8 = 80 \)
Now let's calculate the total number of participants:
\[
5y + 10s = 30 + 80 = 110
\]
Now we check the inequality:
\[
110 \leq 200
\]
This statement is **true**; therefore, the scheduled classes do meet the participant's capacity. The fitness center can accommodate additional participants, as the total of 110 is less than the maximum capacity of 200 participants.
1. **Defining Variables:**
- Let \( Y \) represent the number of participants in yoga classes.
- Let \( S \) represent the number of participants in spin classes.
**Creating the Inequality:**
Given the constraints:
- Each yoga class requires 5 participants, so if there are \( y \) yoga classes, the total number of participants in yoga would be \( 5y \).
- Each spin class requires 10 participants, so if there are \( s \) spin classes, the total number of participants in spin would be \( 10s \).
The total number of participants for both yoga and spin classes can be represented as:
\[
5y + 10s \leq 200
\]
2. **Determining if the Scheduled Classes Meet Capacity:**
The center schedules 6 yoga classes and 8 spin classes. We can substitute these values into the inequality created in part 1.
- Number of yoga participants: \( 5y = 5 \times 6 = 30 \)
- Number of spin participants: \( 10s = 10 \times 8 = 80 \)
Now let's calculate the total number of participants:
\[
5y + 10s = 30 + 80 = 110
\]
Now we check the inequality:
\[
110 \leq 200
\]
This statement is **true**; therefore, the scheduled classes do meet the participant's capacity. The fitness center can accommodate additional participants, as the total of 110 is less than the maximum capacity of 200 participants.