Problem 62 - College Algebra with Modeling and Visualization

Question

Maximizing Storage \(\mathbf{A}\) manager wants to buy filing cabinets. Cabinet \(\mathbf{X}\) costs \(\$ 100\), requires 6 square feet of floor space, and holds 8 cubic feet. Cabinet Y costs \(\$ 200,\) requires 8 square feet of floor space, and holds 12 cubic feet. No more than \(\$ 1400\) can be spent, and the office has room for no more than 72 square feet of cabinets. The manager wants the maximum storage capacity within the limits imposed by funds and space. How many of each type of cabinet should be bought?

Step-by-Step Solution

Answer

The manager should buy 12 Cabinet X and 0 Cabinet Y.

Step 1: Define Variables

Let's define the variables required for the problem. Let:- \( x \) be the number of Cabinet X.- \( y \) be the number of Cabinet Y.

Step 2: Formulate Constraints

We have two primary constraints based on the budget and floor space:1. Budget constraint: \( 100x + 200y \leq 1400 \)2. Floor space constraint: \( 6x + 8y \leq 72 \)

Step 3: Objective Function

The manager wants to maximize storage, which will be our objective function:\( 8x + 12y \). Our goal is to find values of \( x \) and \( y \) such that this expression is maximized.

Step 4: Solve Constraints for Intercepts

To solve the problem, determine the intercepts for each constraint on a graph:- For the budget constraint, if \( y = 0 \), then \( 100x = 1400 \) gives \( x = 14 \); if \( x = 0 \), then \( 200y = 1400 \) gives \( y = 7 \).- For the floor space constraint, if \( y = 0 \), then \( 6x = 72 \) gives \( x = 12 \); if \( x = 0 \), then \( 8y = 72 \) gives \( y = 9 \).

Step 5: Graph Constraints

Plot the lines from both constraints on a graph. The feasible region will be a polygon formed by these lines and the axes.

Step 6: Identify Feasible Points

The feasible region's vertices can be evaluated as potential candidates for maximizing the storage function: - (0,7), - (6,3), - (12,0), - (4,5) - Figure out these from where lines intersect or intercept with axis.

Step 7: Evaluate Objective Function at Vertices

Calculate the value of the storage function \( 8x + 12y \) at each vertex:- At (0,7): \( 8(0) + 12(7) = 84 \)- At (6,3): \( 8(6) + 12(3) = 84 \)- At (12,0): \( 8(12) + 12(0) = 96 \)- At (4,5): \( 8(4) + 12(5) = 92 \)

Step 8: Choose the Maximum

From the calculations, the maximum storage value is 96, achieved at the point (12,0). This point is within the constraints and thus is the optimal solution.

Key Concepts

Objective Function

In linear programming, the objective function is a mathematical expression that we aim to maximize or minimize. In this exercise, it represents the storage capacity that the manager wants to maximize. The function is derived from the contribution of each cabinet type to the total storage. Cabinet X holds 8 cubic feet, and Cabinet Y holds 12 cubics, so our objective function is represented as:\[ 8x + 12y \]This formula indicates that for every Cabinet X purchased, storage increases by 8 cubic feet, and for every Cabinet Y purchased, it increases by 12 cubic feet. The goal is to choose values of \( x \) and \( y \) that maximize the total storage capacity within given limits.

Constraints

Constraints in linear programming define the boundaries within which the solution must be found. They are the limitations or restrictions imposed by the problem. For this scenario, two constraints need to be considered:

• Budget Constraint: The total expenditure on cabinets must not exceed $1400. This is mathematically expressed as: \[ 100x + 200y \leq 1400 \]
• Floor Space Constraint: The cabinets can only occupy up to 72 square feet of space: \[ 6x + 8y \leq 72 \]

These constraints define the feasible region on the graph where any potential solution must lie. Both inequalities represent a real-world limitation on the available resources—namely budget and space.

Feasible Region

The feasible region is a critical concept in linear programming. It represents the set of all possible solutions that satisfy the problem's constraints. In this exercise, the feasible region is the area on the graph where the budget and floor space lines intersect.When plotted, these constraints form a polygonal area where both inequalities hold true. By identifying this region, we can explore its vertices for potential optimal solutions. Each vertex is a possible combination of cabinets \( (x, y) \) that respects both budgetary and space limits.Finding the feasible region's boundaries and vertices helps ensure that we find the optimal amount of storage without surpassing any restrictions.

Maximization Problem

A maximization problem in linear programming seeks to find the highest possible value of the objective function within the constraints. In our task, this involves selecting the number of cabinets X and Y to maximize the storage expressed through the function:\[ 8x + 12y \]Among the points evaluated in the feasible region:

• \((0, 7)\): Yielded 84 cubic feet
• \((6, 3)\): Yielded 84 cubic feet
• \((12, 0)\): Yielded 96 cubic feet
• \((4, 5)\): Yielded 92 cubic feet

The solution is to find the point that results in the maximum value, which is reached at \((12, 0)\). This means buying 12 units of Cabinet X gives the highest possible storage, complying with all constraints.