Answer :
The optimal purchase to maximize storage volume within the given budget and floor space constraints is to buy 6 units of Cabinet X and 3 units of Cabinet Y.
To formulate a linear programming model that maximizes the storage volume, we need to define the decision variables, objective function, and constraints.
Let's define the decision variables:
- Let x be the number of units of Cabinet X.
- Let y be the number of units of Cabinet Y.
Next, we establish the objective function, which is to maximize the storage volume:
- The storage volume for Cabinet X is 8 cubic feet per unit, so for [tex]\( x \)[/tex] units, it is [tex]\( 8x \)[/tex].
- The storage volume for Cabinet Y is 12 cubic feet per unit, so for y units, it is [tex]\( 12y \)[/tex].
- The total storage volume to be maximized is [tex]V\( 8x + 12y \)[/tex].
Now, we set up the constraints:
1. The cost constraint: The total cost of the cabinets should not exceed the budget of $140.
- The cost for Cabinet X is $10 per unit, so for x units, it is 10x .
- The cost for Cabinet Y is $20 per unit, so for y units, it is 20y .
- The cost constraint is [tex]\( 10x + 20y \leq 140 \)[/tex].
2. The floor space constraint: The total floor space occupied by the cabinets should not exceed 72 square feet.
- Cabinet X requires 6 square feet per unit, so for x units, it is 6x.
- Cabinet Y requires 8 square feet per unit, so for y units, it is 8y .
- The floor space constraint is [tex]\( 6x + 8y \leq 72 \)[/tex].
3. Non-negativity constraints: The number of cabinets must be non-negative.
[tex]- \( x \geq 0 \)\\ - \( y \geq 0 \)[/tex]
To solve this linear programming problem, we can use the graphical method or a software tool. Graphically, we would plot the constraints on a coordinate plane and find the feasible region, which is the area where all constraints are satisfied. The optimal solution will be at one of the corner points of this feasible region.
By solving these equations, we find that the maximum storage volume is achieved when x = 6 and y = 3 . This gives us a total storage volume of [tex]\( 8 \times 6 + 12 \times 3 = 48 + 36 = 84 \)[/tex] cubic feet. The total cost for this combination is [tex]\( 10 \times 6 + 20 \times 3 = 60 + 60 = 120 \)[/tex] dollars, which is within the budget. The total floor space used is [tex]\( 6 \times 6 + 8 \times 3 = 36 + 24 = 60 \)[/tex] square feet, which is also within the limit.
Therefore, the optimal purchase is 6 units of Cabinet X and 3 units of Cabinet Y, which maximizes the storage volume while satisfying the budget and floor space constraints."
Final answer:
The linear programming model would be maximizing Z = 8X + 12Y (storage volume for cabinet X and Y) subject to the constraints 10X + 20Y ≤ 140 (budget) and 6X + 8Y ≤ 72 (floor space), with X and Y nonnegative. This would be resolved by graphing the inequalities.
Explanation:
The linear programming model would involve determining the amount of cabinet X and Y to purchase in order to maximize storage capacity, within the constraints of the budget and floor space. We can designate the quantity of cabinet X as X and cabinet Y as Y. The objective function to maximize would be Z = 8X + 12Y, representing the storage volume. Two constraints would include the budget (10X + 20Y ≤ 140) and the space (6X + 8Y ≤ 72). The variables X and Y must be nonnegative. Graph the inequalities to find the feasible region and identify the vertices, applying the objective function to find the maximum storage.
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