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) A
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. Create a two-variable linear inequality that represents the constraint of total number of participants. Clearly define your variables (5 points) A
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)
Answers
GPT-4o mini
1. **Defining the Variables:**
Let:
- \( Y \) = number of yoga classes
- \( S \) = number of spin classes
To represent the constraint of total number of participants, we first need to calculate the number of participants per class type:
- Each yoga class requires 5 participants, so the total participants from yoga classes is \( 5Y \).
- Each spin class requires 10 participants, so the total participants from spin classes is \( 10S \).
The fitness center can accommodate a maximum of 200 participants. Therefore, the inequality representing this constraint is:
\[
5Y + 10S \leq 200
\]
2. **Determining if the Schedule Meets Capacity:**
The center schedules 6 yoga classes and 8 spin classes. We can substitute \( Y = 6 \) and \( S = 8 \) into the inequality:
\[
5(6) + 10(8) \leq 200
\]
Calculating the left side:
\[
30 + 80 = 110
\]
Now we compare it with the maximum capacity:
\[
110 \leq 200
\]
Since this inequality is true, the scheduled classes meet the participant's capacity.
Let:
- \( Y \) = number of yoga classes
- \( S \) = number of spin classes
To represent the constraint of total number of participants, we first need to calculate the number of participants per class type:
- Each yoga class requires 5 participants, so the total participants from yoga classes is \( 5Y \).
- Each spin class requires 10 participants, so the total participants from spin classes is \( 10S \).
The fitness center can accommodate a maximum of 200 participants. Therefore, the inequality representing this constraint is:
\[
5Y + 10S \leq 200
\]
2. **Determining if the Schedule Meets Capacity:**
The center schedules 6 yoga classes and 8 spin classes. We can substitute \( Y = 6 \) and \( S = 8 \) into the inequality:
\[
5(6) + 10(8) \leq 200
\]
Calculating the left side:
\[
30 + 80 = 110
\]
Now we compare it with the maximum capacity:
\[
110 \leq 200
\]
Since this inequality is true, the scheduled classes meet the participant's capacity.
A local theater is planning to host a play. The theater has 300 seats available, and it sells two types of tickets: standard tickets (S) and premium tickets (P). The theater wants to find a combination of scket sales that exactly fills the theater. Each standard ticket sold allows for 1 attendee, while each premium ticket sold allows for 2 attendees
Part 1: Create a two-variable linear equation that represents the total number of atendees when all the seats are filed. Clearly define your variables. (5 Points)
Part 2: The theater sells 100 standard tickets and 100 premium tickets. Determine if this ticket sales decision exactly fills the theater. Show your work and state whether the theater is filled with this given scenario. (2 points)
(7 points)
Part 1: Create a two-variable linear equation that represents the total number of atendees when all the seats are filed. Clearly define your variables. (5 Points)
Part 2: The theater sells 100 standard tickets and 100 premium tickets. Determine if this ticket sales decision exactly fills the theater. Show your work and state whether the theater is filled with this given scenario. (2 points)
(7 points)
GPT-4o mini
### Part 1: Creating a Two-Variable Linear Equation
**Defining the Variables:**
Let:
- \( S \) = number of standard tickets sold
- \( P \) = number of premium tickets sold
Since each standard ticket allows for 1 attendee and each premium ticket allows for 2 attendees, the total number of attendees can be represented by the equation:
\[
S + 2P = 300
\]
This equation states that the sum of attendees from standard tickets and double the attendees from premium tickets will equal the total number of available seats (300).
---
### Part 2: Determining if Ticket Sales Fill the Theater
The theater sells 100 standard tickets and 100 premium tickets. We can substitute \( S = 100 \) and \( P = 100 \) into the equation:
\[
100 + 2(100) = 300
\]
Calculating the left side:
\[
100 + 200 = 300
\]
Now we compare it with the total number of seats:
\[
300 = 300
\]
Since this equation holds true, the theater is exactly filled with this ticket sales decision.
**Conclusion:** The theater is filled with 100 standard tickets and 100 premium tickets.
**Defining the Variables:**
Let:
- \( S \) = number of standard tickets sold
- \( P \) = number of premium tickets sold
Since each standard ticket allows for 1 attendee and each premium ticket allows for 2 attendees, the total number of attendees can be represented by the equation:
\[
S + 2P = 300
\]
This equation states that the sum of attendees from standard tickets and double the attendees from premium tickets will equal the total number of available seats (300).
---
### Part 2: Determining if Ticket Sales Fill the Theater
The theater sells 100 standard tickets and 100 premium tickets. We can substitute \( S = 100 \) and \( P = 100 \) into the equation:
\[
100 + 2(100) = 300
\]
Calculating the left side:
\[
100 + 200 = 300
\]
Now we compare it with the total number of seats:
\[
300 = 300
\]
Since this equation holds true, the theater is exactly filled with this ticket sales decision.
**Conclusion:** The theater is filled with 100 standard tickets and 100 premium tickets.