Question

Cabinet X costs ​$100​, requires 15sq ft of floor​ space, and holds 30cu ft of files. Cabinet Y costs ​$120​, requires 6sq ft of floor​ space, and holds 30 cu ft. To get maximum​ storage, how many of each should be purchased with a budget limit of ​$1200 and floor space of 200sq​ ft?

Answer 1

The optimal solution to the problem is to buy 8 Cabinet X and 2 Cabinet Y to achieve the maximum storage with the given limitations.

Step-by-step explanation:

The student is dealing with a linear programming problem that requires optimizing two variables with given constraints. The objective is to maximize file storage by purchasing cabinets with a budget of $1200 and a limit of 200 sq ft of floor space. To solve this, we can use a system of inequalities.

Let x be the number of Cabinet X and y be the number of Cabinet Y. The cost constraint is given as 100x + 120y ≤ 1200, and the floor space constraint is as 15x + 6y ≤ 200. Since we cannot have negative numbers of cabinets, we also have the constraints x ≥ 0 and y ≥ 0.

First, we find the maximum number of each cabinet that can be bought without violating the individual constraints:

For Cabinet X, with cost constraint: 100x ≤ 1200 → x ≤ 12.

For Cabinet Y, with floor space constraint: 6y ≤ 200 → y ≤ 33.½, but since we can't have half a cabinet, y ≤ 33.

Next, we plot the constraints on a graph, with x on the horizontal axis and y on the vertical axis to find the feasible region and thus, the possible combinations.

Finally, we test the vertices of the feasible region to determine the combination that allows for the maximum number of cabinets while staying within budget and space constraints.

After solving this, we find that the optimal purchase combination to get the maximum storage is buying 8 Cabinet X and 2 Cabinet Y, using all the available budget and floor space.