Imperial Units and Floor Space
3.4 Linear Programming
Linear programming is one of the most useful types of word problems we learn in Algebra 2. It takes a collection of information and produces the best possible solution. However, each problem is unique. This requires an overall understanding of how to set these up.
You need to buy some filing cabinets. You know that Cabinet X costs $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y costs $20 per unit, requires eight square feet of floor space, and holds twelve cubic feet of files. You have been given $140 for this purchase, though you don't 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?
The question asks for the number of cabinets I need to buy, so my variables will stand for that: x: number of model X cabinets purchased y: number of model Y cabinets purchased Separate all the information. This is optional but helpful. Cabinet X Cabinet Y Restrictions $10 per unit $20 per unit $140 6 ft2 floor space 8 ft2 floor space 72 ft2 floor space 8 ft3 storage 12 ft3 storage (maximum)
Set up your equations. They practically fall right out of the information above.
x > 0 and y > 0 are always two of the constraints because you cannot have negative filing cabinets. cost: 10x + 20y < 140 y < – 1/2 x + 7 (for graphing) space: 6x + 8y < 72 y < – 3/4 x + 9 (for graphing)
Determine the optimizing equation. Looking at the table, it is the row without a restriction. In this case, it is the volume of the filing cabinets. volume: V = 8x + 12y
Graph the inequalities. This system (along with the first two constraints) graphs as:
List the corner points and evaluate them into the optimizing equation. (8, 3) = 8(8) + 12 (3) = 100 ft3 (0, 7) = 8(0) + 12(7) = 84 ft3 (12, 0) =
Linear programming is one of the most useful types of word problems we learn in Algebra 2. It takes a collection of information and produces the best possible solution. However, each problem is unique. This requires an overall understanding of how to set these up.
You need to buy some filing cabinets. You know that Cabinet X costs $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y costs $20 per unit, requires eight square feet of floor space, and holds twelve cubic feet of files. You have been given $140 for this purchase, though you don't 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?
The question asks for the number of cabinets I need to buy, so my variables will stand for that: x: number of model X cabinets purchased y: number of model Y cabinets purchased Separate all the information. This is optional but helpful. Cabinet X Cabinet Y Restrictions $10 per unit $20 per unit $140 6 ft2 floor space 8 ft2 floor space 72 ft2 floor space 8 ft3 storage 12 ft3 storage (maximum)
Set up your equations. They practically fall right out of the information above.
x > 0 and y > 0 are always two of the constraints because you cannot have negative filing cabinets. cost: 10x + 20y < 140 y < – 1/2 x + 7 (for graphing) space: 6x + 8y < 72 y < – 3/4 x + 9 (for graphing)
Determine the optimizing equation. Looking at the table, it is the row without a restriction. In this case, it is the volume of the filing cabinets. volume: V = 8x + 12y
Graph the inequalities. This system (along with the first two constraints) graphs as:
List the corner points and evaluate them into the optimizing equation. (8, 3) = 8(8) + 12 (3) = 100 ft3 (0, 7) = 8(0) + 12(7) = 84 ft3 (12, 0) =