It looks like you're outlining a scenario for a fitness center's group class promotion with certain constraints regarding class sizes and requirements. Based on the information provided, here's how we can describe and analyze the scenario mathematically:
Variables:
- Let \( y \) be the number of yoga classes offered.
- Let \( s \) be the number of spin classes offered.
Constraints:
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Maximum Capacity: The total number of participants cannot exceed 200. \[ 5y + 10s \leq 200 \]
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Minimum Enrollment per Class: Each class requires a minimum number of participants:
- Yoga needs at least 5 participants per class.
- Spin needs at least 10 participants per class.
This means:
- For Yoga: \( y \geq 0 \) (any non-negative number since you can't have negative classes).
- For Spin: \( s \geq 0 \) (same as above).
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Feasibility of Classes: If there are classes, they must have the required participants:
- If there is a yoga class, then \( y \) must be such that \( 5y \) ≤ total number of participants.
- If there is a spin class, then \( s \) must be such that \( 10s \) ≤ total number of participants.
Objective:
If there's a specific goal for the promotion (e.g., maximizing participation or revenue), we can establish an objective function.
Example Objective Function:
Assuming you want to maximize the number of classes while adhering to the constraints:
- Maximize \( y + s \) (the total number of classes).
Solving:
To determine feasible combinations of yoga and spin classes, you can rearrange the inequality:
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Start with the capacity constraint: \[ 5y + 10s \leq 200 \] Simplifying this gives: \[ y + 2s \leq 40 \] This implies that the maximum combinations of yoga and spin classes should be such that their weighted sum (with yoga classes contributing half the weight of a spin) does not exceed 40.
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From this point, you can test different values for \( s \) and calculate the corresponding \( y \).
Example Values:
- If \( s = 0 \): \[ y + 2(0) \leq 40 \implies y \leq 40 \]
- If \( s = 10 \): \[ y + 2(10) \leq 40 \implies y \leq 20 \]
- If \( s = 20 \): \[ y + 2(20) \leq 40 \implies y \leq 0 \]
Putting this together gives a feasible region for combinations of yoga and spin classes based on the participant constraints.
Conclusion:
You can use this framework to determine the best class offering arrangement under the mentioned constraints for your fitness center promotion. If you have any specific aspects to discuss further, feel free to elaborate!