A lawn and garden store creates three different potting mixes sold in​ 20-pound bags. Mix A contains 12 pounds of​ peat, 5 pounds of​ perlite, 3 pounds of organic​ material, and earns a profit of ​$3 per bag. Mix B contains 10 pounds of​ peat, 6 pounds of​ perlite, 4 pounds of organic​ material, and earns a profit of​ $5 per bag. Mix C contains 8 pounds of​ peat, 8 pounds of​ perlite, 4 pounds of organic​ material, and earns a profit of​ $6 per bag. There are 1 comma 216 pounds of​ peat, 1 comma 040 pounds of​ perlite, and 720 pounds of organic material available every week to make these potting mixes. For what values of h will a sensitivity analysis on the effect of a change of h pounds of peat be​ valid?
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Part 1
For what values of h will a sensitivity analysis on the effect of a change of h pounds of peat be​ valid?
  
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​(Simplify your answers. Type integers or simplified​ fractions.)

To conduct a sensitivity analysis, we need to determine the range of values for $ h $ that would not violate the availability of resources, in this case, peat. Here's how to proceed with solving the problem:

### Step 1: Define the problem and constraints

The available quantities of materials (peat, perlite, organic material) set the constraints. Specifically, the availability of peat is 1,216 pounds per week.

Let $ x_A $, $ x_B $, and $ x_C $ represent the number of bags of mix A, mix B, and mix C produced, respectively.

- **Peat constraint**: 
   - Mix A uses 12 pounds of peat per bag, so 12$x_A$ pounds for $ x_A $ bags.
   - Mix B uses 10 pounds of peat per bag, so 10$x_B$ pounds for $ x_B $ bags.
   - Mix C uses 8 pounds of peat per bag, so 8$x_C$ pounds for $ x_C $ bags.

The total amount of peat used must be less than or equal to the available amount of peat, which is $ 1216 + h $ pounds (where $ h $ is the extra pounds of peat). Thus, the constraint is:

\[
12x_A + 10x_B + 8x_C \leq 1216 + h
\]

### Step 2: Set up the sensitivity analysis constraint

We want to know the range of $ h $ (pounds of extra peat) for which this analysis is valid. For the sensitivity analysis to remain valid, $ h $ must keep the resource constraint feasible without exceeding any limits.

Thus, the change in $ h $ must be non-negative (since increasing $ h $ means increasing the availability of peat). Therefore:

\[
0 \leq h \leq 1216
\]

The upper limit is defined by the total amount of peat available (initial 1216 pounds) and any increase $ h $, meaning the availability must remain positive.

### Final answer:

The sensitivity analysis will be valid for $ 0 \leq h \leq 1216 $.

Do you want more details or have any questions? 

Here are 5 relative questions to extend this topic:
1. How would a sensitivity analysis differ if there were constraints on perlite or organic material instead?
2. How does changing $ h $ affect the total profit made from the different mixes?
3. What happens to the solution if the profit per bag changes for one of the mixes?
4. How would you graph the constraint equations for different values of $ h $?
5. What are the implications of reducing the availability of organic material by a similar amount?

**Tip**: Sensitivity analysis helps determine how changes in resource availability or parameters affect the outcome of an optimization problem.

A lawn and garden store creates three different potting mixes sold in 20-pound bags. Mix A contains 12 pounds of peat, 5 pounds of perlite, 3 pounds of organic material, and earns a profit of $3 per bag. Mix B contains 10 pounds of peat, 6 pounds of perlite, 4 pounds of organic material, and earns a profit of $5 per bag. Mix C contains 8 pounds of peat, 8 pounds of perlite, 4 pounds of organic material, and earns a profit of $6 per bag. There are 1,216 pounds of peat, 1,040 pounds of perlite, and 720 pounds of organic material available every week to make these potting mixes. For what values of h will a sensitivity analysis on the effect of a change of h pounds of peat be valid?

Explore how to conduct a sensitivity analysis for a potting mix problem using linear programming. Learn to set up constraints for peat, perlite, and organic materials and determine the range of values for h (pounds of peat) that ensure feasibility. Perfect for students learning optimization and resource management.

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