Worksheet: Linear Programming

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In this worksheet, we will practice finding the optimal solution of a linear system that has an objective function and multiple constraints.

Q1:

Determine the values of 𝑥 and 𝑦 that maximize the function 𝑝=5𝑥+2𝑦. Write your answer as a point (𝑥,𝑦).

  • A ( 0 , 8 )
  • B ( 7 , 8 )
  • C ( 7 , 0 )
  • D ( 3 , 0 )

Q2:

Find the maximum value of the objective function 𝑝=2𝑥+6𝑦 given the constraints 𝑥≥0, 𝑦≥0, 𝑥+𝑦≤6, 3𝑥+𝑦≤9, and 𝑥+2𝑦≤8.

Q3:

Given the graph below and that 𝑥≥0, 𝑦≥0, 𝑥+𝑦≤7, and 𝑦≥5, determine at which point the function 𝑝=3𝑥−𝑦 has its maximum using linear programming.

  • A 𝐴
  • B 𝐷
  • C 𝐶
  • D 𝐵

Q4:

Using linear programming, find the minimum and maximum values of the function 𝑝=4𝑥−3𝑦 given that 𝑥≥0, 𝑦≥0, 𝑥+𝑦≤9, and 𝑦≥5 .

  • AThe minimum value is −27, and the maximum value is 1.
  • BThe minimum value is −15, and the maximum value is 1.
  • CThe minimum value is −27, and the maximum value is −15.
  • DThe minimum value is 0, and the maximum value is 9.

Q5:

Minimize 𝑧=𝑥+𝑥 subject to the constraints 𝑥+𝑥≥2, 𝑥+3𝑥≤20, and 𝑥+𝑥≤18.

  • A1
  • B7
  • C2
  • D18
  • E9

Q6:

Consider the following inequalities in the nonnegative variables 𝑥, 𝑥, and 𝑥: 𝑥+𝑥+𝑥≤10,𝑥+𝑥+𝑥≥2,𝑥+2𝑥+𝑥≤7. Find the maximum and minimum possible values of 𝑧=𝑥−2𝑥+𝑥 subject to these constraints.

  • Aminimum: −7, maximum: 7
  • Bminimum: −7, maximum: 5
  • Cminimum: −132, maximum: 5
  • Dminimum: −72, maximum: 7
  • Eminimum: −112, maximum: 5

Q7:

Consider the following inequalities in the nonnegative variables 𝑥, 𝑥, and 𝑥: 𝑥+𝑥+𝑥≤8,𝑥+𝑥+3𝑥≥1,𝑥+𝑥+𝑥≤7. Find the maximum and minimum possible values of 𝑧=𝑥−2𝑥−3𝑥 subject to these constraints.

  • Aminimum: −21, maximum: 6
  • Bminimum: −20, maximum: 6
  • Cminimum: −21, maximum: 7
  • Dminimum: −20, maximum: 7

Q8:

Consider the following inequalities in the nonnegative variables 𝑥, 𝑥, and 𝑥: 𝑥−𝑥+𝑥≤10,𝑥+𝑥+𝑥≥1,𝑥+2𝑥+𝑥≤7. Find the maximum and minimum possible values of 𝑧=2𝑥+𝑥 subject to these constraints.

  • Aminimum: 1, maximum: 14
  • Bminimum: 1, maximum: 7
  • Cminimum: 0, maximum: 14
  • Dminimum: 0, maximum: 7

Q9:

Consider the following inequalities in the nonnegative variables 𝑥, 𝑥, and 𝑥: 𝑥+𝑥+𝑥≤10,𝑥+𝑥+𝑥≥1,𝑥+2𝑥+𝑥≤7. Find the maximum and minimum possible values of 𝑧=𝑥−2𝑥 subject to these constraints.

  • Aminimum: −72, maximum: 7
  • Bminimum: −7, maximum: 6
  • Cminimum: −132, maximum: 6
  • Dminimum: −7, maximum: 7

Q10:

Consider the following inequalities in the nonnegative variables 𝑥, 𝑥, and 𝑥: 𝑥−𝑥+𝑥≤10,𝑥+𝑥+𝑥≥1,𝑥+2𝑥+𝑥≤7. Find the maximum and minimum possible values of 𝑧=𝑥+2𝑥 subject to these constraints.

  • Aminimum: 0, maximum: 272
  • Bminimum: 0, maximum: 7
  • Cminimum: 1, maximum: 272
  • Dminimum: 1, maximum: 7

Q11:

A seafood restaurant sells two types of cooked fish; cod and eel. The restaurant sells NO LESS than 40 fish every day but it does not use more than 30 cod and no more than 45 eels. The price of one cod is 6 LE and that of an eel is 8 LE. Let 𝑥 represent the amount of cod purchased each day, and 𝑦 represent the amount of eel. Given that the manager wants to minimise the total price, 𝑝, of fish, state the objective function and the inequalities that will help the restaurant manager decide how many of each fish to buy.

  • A 𝑥 ≥ 0 , 𝑦 ≥ 0 , 𝑥 + 𝑦 ≥ 4 0 , 𝑥 < 3 0 , 𝑦 < 4 5 , 𝑝 = 6 𝑥 + 8 𝑦
  • B 𝑥 ≥ 0 , 𝑦 ≥ 0 , 𝑥 + 𝑦 > 4 0 , 𝑥 ≤ 3 0 , 𝑦 ≤ 4 5 , 𝑝 = 6 𝑥 + 8 𝑦
  • C 𝑥 ≥ 0 , 𝑦 ≥ 0 , 𝑥 + 𝑦 ≥ 4 0 , 𝑥 ≥ 3 0 , 𝑦 ≥ 4 5 , 𝑝 = 6 𝑥 + 8 𝑦
  • D 𝑥 ≥ 0 , 𝑦 ≥ 0 , 𝑥 + 𝑦 ≥ 4 0 , 𝑥 ≤ 3 0 , 𝑦 ≤ 4 5 , 𝑝 > 6 𝑥 + 8 𝑦
  • E 𝑥 ≥ 0 , 𝑦 ≥ 0 , 𝑥 + 𝑦 ≥ 4 0 , 𝑥 ≤ 3 0 , 𝑦 ≤ 4 5 , 𝑝 = 6 𝑥 + 8 𝑦

Q12:

A candy store sells bags of marshmallows for 5 LE each and bags of cola candy for 6 LE each. A child wants to buy both types of candy and has restrictions on how many they can buy that are described by the figure shown, where 𝑥 represents the number of bags of marshmallows they buy and 𝑦 represents the number of bags of cola candy. What is the lowest price possible in this situation?

Q13:

A small factory produces two types of metal furniture, 𝐴 and 𝐵. They can produce at most 25 pieces of metal furniture in total. The profit from type 𝐴 is 60 LE and the profit from type 𝐵 is 40 LE. The factory sells at least 2 times more of type 𝐴 than type 𝐵. State the objective function and inequalities that will help to find the maximum profit for the factory.

  • A 𝐴 ≥ 0 , 𝐵 ≥ 0 , 𝐴 + 𝐵 ≥ 2 5 , 𝐴 = 2 𝐵 , 𝑝 = 6 0 𝐴 + 4 0 𝐵
  • B 𝐴 ≥ 0 , 𝐵 ≥ 0 , 𝐴 + 𝐵 ≤ 2 5 , 𝐴 ≥ 2 𝐵 , 𝑝 ≥ 6 0 𝐴 + 4 0 𝐵
  • C 𝐴 ≥ 0 , 𝐵 ≥ 0 , 𝐴 + 𝐵 ≤ 2 5 , 𝐴 ≥ 2 𝐵 , 𝑝 = 6 0 𝐴 + 4 0 𝐵
  • D 𝐴 ≥ 0 , 𝐵 ≥ 0 , 𝐴 + 𝐵 = 2 5 , 𝐴 ≥ 2 𝐵 , 𝑝 = 6 0 𝐴 + 4 0 𝐵

Q14:

A baby food factory produces two types of baby food with different nutritional values. The first type, denoted by 𝑥, costs 3 LE for one jar which contains 3 units of vitamin A and 2 of vitamin B. The second type, denoted by 𝑦, costs 4 LE for one jar which contains 4 units of vitamin A and 3 of vitamin B. A child needs at least 120 units of vitamin A and 100 units of vitamin B to satisfy his nutrition needs. State the objective function and the inequality constraints needed to determine how many jars of each type to purchase to satisfy the nutrition requirements at the lowest possible cost.

  • A 𝑥 ≥ 0 , 𝑦 ≥ 0 , 3 𝑥 + 4 𝑦 ≥ 1 2 0 , 2 𝑥 + 3 𝑦 ≥ 1 0 0 , 𝑝 = 3 𝑥 + 4 𝑦
  • B 𝑥 ≥ 0 , 𝑦 ≥ 0 , 3 𝑥 + 2 𝑦 ≥ 1 2 0 , 4 𝑥 + 3 𝑦 ≥ 1 0 0 , 𝑝 = 3 𝑥 + 4 𝑦
  • C 𝑥 ≥ 0 , 𝑦 ≥ 0 , 3 𝑥 + 4 𝑦 ≥ 1 2 0 , 2 𝑥 + 3 𝑦 ≥ 1 0 0 , 𝑝 ≥ 3 𝑥 + 4 𝑦
  • D 𝑥 ≥ 0 , 𝑦 ≥ 0 , 3 𝑥 + 4 𝑦 ≤ 1 2 0 , 2 𝑥 + 3 𝑦 ≤ 1 0 0 , 𝑝 = 3 𝑥 + 4 𝑦

Q15:

A baby food factory produces two types of baby food with different nutritional values. One jar of the first type has 2 units of vitamin A and 4 units of vitamin B, while a jar of the second type has 4 units of vitamin A and 2 units of vitamin B. Every child requires at least 100 units of vitamin A and 140 units of vitamin B each month. The first type costs 6 LE per jar while the second costs 4 LE per jar. Using the graph below, determine the objective function and then find the lowest possible cost required to supply a child with their required monthly nutrients.

  • A 𝑝 = 4 𝑥 + 6 𝑦 , and the lowest possible cost is 420 LE.
  • B 𝑝 = 6 𝑥 + 4 𝑦 , and the lowest possible cost is 280 LE.
  • C 𝑝 = 2 𝑥 + 4 𝑦 , and the lowest possible cost is 100 LE.
  • D 𝑝 = 4 𝑥 + 6 𝑦 , and the lowest possible cost is 1,800 LE.
  • E 𝑝 = 6 𝑥 + 4 𝑦 , and the lowest possible cost is 220 LE.

Q16:

A baby food factory produces two types of baby food with different nutritional values. One jar of the first type has 4 units of vitamin A and 2 units of vitamin B, while a jar of the second type has 2 units of vitamin A and 3 units of vitamin B. Every child requires at least 120 units of vitamin A and 100 units of vitamin B each month. The first type costs 6 LE per jar while the second costs 4 LE per jar. Using the graph below, determine how many of each type of jar should be bought to meet the child’s monthly needs at the lowest possible cost.

  • Ajars of the first type =30, jars of the second type =0
  • Bjars of the first type =20, jars of the second type =20
  • Cjars of the first type =0, jars of the second type =33
  • Djars of the first type =0, jars of the second type =60

Q17:

In a workshop, two workers produce two types of iron desks: type A and type B. One worker builds the desks and the other sprays them. It takes the first worker 4 hours to build one desk of type A and 3 hours to build one desk of type B. It takes the second worker 3 hours to spray one desk of type A and 4 hours to spray one desk of type B. The first person works at least 5 hours a day, and the other works a maximum of 7 hours a day. If the workshop earns a profit of 60 LE from each desk (of either type), determine the objective function and inequalities required for calculating the number of desks of each type to be produced every day to maximize the profit 𝑝.

  • A 𝑥 ≥ 0 , 𝑦 ≥ 0 , 4 𝑥 + 3 𝑦 > 5 , 3 𝑥 + 4 𝑦 < 7 , 𝑝 ≥ 6 0 𝑥 + 6 0 𝑦
  • B 𝑥 ≥ 0 , 𝑦 ≥ 0 , 4 𝑥 + 3 𝑦 ≤ 5 , 3 𝑥 + 4 𝑦 ≥ 7 , 𝑝 = 6 0 𝑥 + 6 0 𝑦
  • C 𝑥 ≥ 0 , 𝑦 ≥ 0 , 4 𝑥 + 3 𝑦 ≥ 5 , 3 𝑥 + 4 𝑦 ≤ 7 , 𝑝 = 6 0 𝑥 + 6 0 𝑦
  • D 𝑥 ≥ 0 , 𝑦 ≥ 0 , 4 𝑥 + 3 𝑦 ≥ 5 , 3 𝑥 + 4 𝑦 ≥ 7 , 𝑝 ≥ 6 0 𝑥 + 6 0 𝑦
  • E 𝑥 ≥ 0 , 𝑦 ≥ 0 , 4 𝑥 + 3 𝑦 < 5 , 3 𝑥 + 4 𝑦 > 7 , 𝑝 = 6 0 𝑥 + 6 0 𝑦

Q18:

A factory produces two types of iron desks: Type A and Type B. One worker builds the desks and another sprays them. It takes the first worker 3.5 hours to build one desk of type A and 2 hours to build one desk of type B. It takes the second worker 4 hours to spray one desk of type A and 2 hours to spray one desk of type B. The first person works at least 5 hours a day, and the other works a maximum of 8 hours a day. If the factory earns a profit of 50 LE from each desk (of either type), determine how many desks of each type to produce each day to maximize the profit.

  • A0 desks of type A, 4 desks of type B
  • B0 desks of type A, 2 desks of type B
  • C2 desks of type A, 0 desks of type B
  • D4 desks of type A, 0 desks of type B

Q19:

Two packages of food supplies are available; the first gives 4 calories and has 6 units of vitamin C, and the second gives 3 calories and has 4 units of vitamin C. We need at least 37 calories and 22 units of vitamin C. The first costs 6 LE per package, and the second costs 8 LE per package. Using 𝑥 to represent the amount of packages of the first type and 𝑦 to represent the number of packages of the second type, state the objective function used to determine the minimum cost of buying packages to fulfill the nutrient needs.

  • A 4 𝑥 + 3 𝑦 ≥ 3 7
  • B 𝑝 = 6 𝑥 + 8 𝑦
  • C 𝑝 = 6 𝑥 + 4 𝑦
  • D 𝑝 = 3 7 𝑥 + 2 2 𝑦
  • E 𝑝 < 6 𝑥 + 8 𝑦

Q20:

A farmer can improve the quality of his produce if he uses at least 18 units of nitrogen-based compounds and at least 6 units of phosphate compounds. He can use two types of fertilizers: A and B. The cost and contents of each fertilizer are shown in the table.

The Fertilizer Number of Units of Nitrogen-Based Compounds per Kilogram Number of Units of Phosphate Compounds per Kilogram Cost for Each Kilogram (LE)
A 3 2 170
B 6 1 120

Given that the graph represents the constraints in this situation, find the lowest cost the farmer can pay for fertilizer while providing sufficient amounts of both compounds.

Q21:

A factory produces chairs and tables and is trying to decide how many of each it needs to produce to maximize its profit.

They have determined the constraints and drawn the feasible region as shown, where 𝑥 represents the number of chairs and 𝑦 represents the number of tables.

If they find a buyer who agrees to pay a fee such that they receive 150 profit for each chair and 200 profit for each table, what can they expect their maximum profit to be?

If they can only guarantee a profit of 50 per chair and 180 per table, how many of each should they produce to maximize their profit?

  • A32 chairs, 0 tables
  • B45 chairs, 0 tables
  • C38 chairs, 18 tables
  • D18 chairs, 38 tables
  • E0 chairs, 32 tables

Q22:

A small company dyes shirts to be either solid color or tie-dye, and they want to decide how many shirts of each color to prepare for an upcoming sale. They have a budget of $240. Purchasing each shirt costs $2. It costs $0.50 to dye a shirt with a solid color and $1.50 to produce a tie-dye shirt. They only have 8 hours to prepare all the shirts, and it takes 2 minutes to dye a solid-color shirt and 10 minutes to dye a tie-dye shirt.

They want to maximize their profit, knowing that they can sell solid-color shirts for $8 each and tie-dye shirts for $10 each.

Let 𝑥 represent the number of solid-color shirts and 𝑦 represent the number of tie-dye shirts. Which of the following shows the feasible region?

  • A
  • B
  • C
  • D

State the objective function.

  • A 𝑓 ( 𝑥 , 𝑦 ) = 2 𝑥 + 1 . 5 𝑦 + 2 4 0
  • B 𝑓 ( 𝑥 , 𝑦 ) = 2 𝑥 + 1 0 𝑦 − 4 8 0
  • C 𝑓 ( 𝑥 , 𝑦 ) = 8 𝑥 + 1 0 𝑦
  • D 𝑓 ( 𝑥 , 𝑦 ) = 2 . 5 𝑥 + 3 . 5 𝑦 − 2 4 0
  • E 𝑓 ( 𝑥 , 𝑦 ) = 2 𝑥 + 1 . 5 𝑦

How many of each type of shirt should the company produce to maximize profit?

  • A0 solid-color shirts and 48 tie-dye shirts
  • B89 solid-color shirts and 69 tie-dye shirts
  • C69 solid-color shirts and 40 tie-dye shirts
  • D40 solid-color shirts and 40 tie-dye shirts
  • E48 solid-color shirts and 0 tie-dye shirts

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