Graphing problem

Question

You need to buy some filing cabinets. You know cabinet x costs $10 per unit, requires 6 square feet of floor space and holds 8 cubic feet of files. Cabinet y costs $20 per unit, requires 8 square feet of floor space and holds twelve cubic feet of files. you have been given $140 for this purchase, though you don't know have to spend that much. the office has room for no more than 72 square feet of cabinets. how many of which model should you buy in order to maximize storage volume?

Answer 1

This is a linear programming problem, where we want to maximize storage space while staying within the budget and available floor space.

Step 1: Define the variables.

Let:

1. x = number of Model X cabinets
2. y = number of Model Y cabinets

We know:

Model X:

1. Costs $10
2. Uses 6 square feet of floor space
3. Holds 8 cubic feet of files

Model Y:

1. Costs $20
2. Uses 8 square feet of floor space
3. Holds 12 cubic feet of files

Step 2: Write the constraints.

Budget constraint:

10x + 20y ≤ 140

Divide by 10:

x + 2y ≤ 14

Floor space constraint:

6x + 8y ≤ 72

Divide by 2:

3x + 4y ≤ 36

We also know:

x ≥ 0 and y ≥ 0

Both variables must be whole numbers because we cannot purchase fractions of cabinets.

Step 3: Write the objective function.

We want to maximize the total storage volume.

V = 8x + 12y

Step 4: Find the corner points of the feasible region.

The maximum value of a linear objective function occurs at a corner point of the feasible region.

First, find the intercepts of the constraints.

For the budget constraint:

x + 2y = 14

Intercepts: (14, 0) and (0, 7)

For the floor space constraint:

3x + 4y = 36

Intercepts: (12, 0) and (0, 9)

The feasible region is in the first quadrant and below both lines.

Its corner points are:

(0, 0), (12, 0), (8, 3), and (0, 7).

To find the intersection (8, 3), solve the two equations together:

x + 2y = 14

3x + 4y = 36

Multiply the first equation by 3:

3x + 6y = 42

Subtract the second equation:

2y = 6

y = 3

Substitute back:

x + 2(3) = 14

x = 8

Step 5: Evaluate the objective function at each corner point.

V = 8x + 12y

At (0, 0):

V = 0

At (12, 0):

V = 8(12) = 96 cubic feet

At (8, 3):

V = 8(8) + 12(3)

V = 64 + 36 = 100 cubic feet

At (0, 7):

V = 12(7) = 84 cubic feet

The greatest storage volume is 100 cubic feet.

Step 6: Verify the constraints.

Buying 8 Model X cabinets and 3 Model Y cabinets:

Total cost:

10(8) + 20(3) = $140

Total floor space:

6(8) + 8(3) = 72 square feet

Both constraints are satisfied exactly!

Final Answer: Buy 8 Model X cabinets and 3 Model Y cabinets.

This combination provides a maximum storage capacity of 100 cubic feet, uses exactly 72 square feet of floor space, and costs exactly $140.

Since both quantities are whole numbers, the solution is practical and does not require any rounding.