QUANTITATIVE METHODS FINAL EXAM QUESTIONS AND CORRECT
ANSWERS WITH RATIONALES
Linear Programming & Optimization
1. A hot dog manufacturer wishes to minimize the cost in dollars of producing a low-
cost niched product while meeting the dietary guidelines for protein and sodium.
Once the model has been run, the surplus variable in the sodium constraint has a
value of 1300 milligrams. The best interpretation of this outcome is:
Correct Answer: The minimum cost hot dog has 1300 milligrams more sodium than
required.
Rationale: A surplus variable in a ≥ constraint represents the amount by which the
constraint is exceeded. Since sodium is likely a minimum requirement (≥ constraint),
a surplus of 1300 mg means the solution exceeds the minimum sodium requirement
by 1300 mg.
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2. A model is a functional relationship that includes:
Correct Answer: All of the above (equations, variables, and parameters)
Rationale: A complete mathematical model consists of equations that relate
variables (unknowns we solve for) and parameters (known constants that define the
specific problem context).
,---
3. A slack variable:
Correct Answer: is the amount by which the left side of a ≤ constraint is smaller
than the right side.
Rationale: Slack variables are added to ≤ constraints to convert them to
equations. They represent unused resources. For example, if 2x + 3y ≤ 100, the
slack = 100 - (2x + 3y).
---
4. A university is planning a seminar. It costs $3000 to reserve a room, hire an
instructor, and bring in the equipment. Assume it costs $25 per student for the
administrators to provide the course materials. If we know that 20 people will
attend, what price should be charged per person to break even?
Correct Answer: $175
Rationale: Total Cost = Fixed Cost + Variable Cost × Quantity = $3000 +
$25(20) = $3500. Break-even means Total Revenue = Total Cost, so Price × 20 =
$3500, Price = $175.
---
5. Administrators at a university will charge students $150 to attend a seminar. It
costs $3000 to reserve a room, hire an instructor, and bring in the equipment.
Assume it costs $25 per student for the administrators to provide the course
,materials. How many students would have to register for the seminar for the
university to break even?
Correct Answer: 24
Rationale: Break-even quantity = Fixed Cost ÷ (Price - Variable Cost) = $3000 ÷
($150 - $25) = $3000 ÷ $125 = 24 students.
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6. At the break-even point:
Correct Answer: total revenue equals total cost
Rationale: The break-even point is defined as the quantity where total revenue
equals total cost, resulting in zero profit.
---
7. Belmont has bids from two companies to build its new indoor sports facility. It
will choose between Company A and Company B. Company A has a fixed cost of
75,000 and a variable cost of 15. Company B's fixed cost is 90,000 and variable
cost is 11. Use the total cost equation for each company and compare them. At
what volume is the total cost the same, that is at what volume is Belmont indifferent
between the two bidders?
Correct Answer: 3,750
, Rationale: Set Total Cost A = Total Cost B: 75,000 + 15Q = 90,000 + 11Q →
4Q = 15,000 → Q = 3,750.
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8. Cully Furniture buys two products for resale: big shelves (B) and medium
shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage
space, and each medium shelf costs $300 and requires 90 cubic feet of storage
space. The company has $75,000 to invest in shelves this week, and the warehouse
has 18,000 cubic feet available for storage. Profit for each big shelf is $300 and
for each medium shelf is $150. What is the maximum profit?
Correct Answer: $45,000
Rationale: This requires solving the LP: Max Z = 300B + 150M subject to 500B +
300M ≤ 75,000 and 100B + 90M ≤ 18,000. The optimal solution is B = 150, M =
0, giving Z = $45,000.
---
9. Cully Furniture buys two products for resale: big shelves (B) and medium
shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage
space, and each medium shelf costs $300 and requires 90 cubic feet of storage
space. The company has $75,000 to invest in shelves this week, and the warehouse
has 18,000 cubic feet available for storage. Profit for each big shelf is $300 and
for each medium shelf is $150. In order to maximize profit, how many big shelves
(B) and how many medium shelves (M) should be purchased?
Correct Answer: B = 150, M = 0
ANSWERS WITH RATIONALES
Linear Programming & Optimization
1. A hot dog manufacturer wishes to minimize the cost in dollars of producing a low-
cost niched product while meeting the dietary guidelines for protein and sodium.
Once the model has been run, the surplus variable in the sodium constraint has a
value of 1300 milligrams. The best interpretation of this outcome is:
Correct Answer: The minimum cost hot dog has 1300 milligrams more sodium than
required.
Rationale: A surplus variable in a ≥ constraint represents the amount by which the
constraint is exceeded. Since sodium is likely a minimum requirement (≥ constraint),
a surplus of 1300 mg means the solution exceeds the minimum sodium requirement
by 1300 mg.
---
2. A model is a functional relationship that includes:
Correct Answer: All of the above (equations, variables, and parameters)
Rationale: A complete mathematical model consists of equations that relate
variables (unknowns we solve for) and parameters (known constants that define the
specific problem context).
,---
3. A slack variable:
Correct Answer: is the amount by which the left side of a ≤ constraint is smaller
than the right side.
Rationale: Slack variables are added to ≤ constraints to convert them to
equations. They represent unused resources. For example, if 2x + 3y ≤ 100, the
slack = 100 - (2x + 3y).
---
4. A university is planning a seminar. It costs $3000 to reserve a room, hire an
instructor, and bring in the equipment. Assume it costs $25 per student for the
administrators to provide the course materials. If we know that 20 people will
attend, what price should be charged per person to break even?
Correct Answer: $175
Rationale: Total Cost = Fixed Cost + Variable Cost × Quantity = $3000 +
$25(20) = $3500. Break-even means Total Revenue = Total Cost, so Price × 20 =
$3500, Price = $175.
---
5. Administrators at a university will charge students $150 to attend a seminar. It
costs $3000 to reserve a room, hire an instructor, and bring in the equipment.
Assume it costs $25 per student for the administrators to provide the course
,materials. How many students would have to register for the seminar for the
university to break even?
Correct Answer: 24
Rationale: Break-even quantity = Fixed Cost ÷ (Price - Variable Cost) = $3000 ÷
($150 - $25) = $3000 ÷ $125 = 24 students.
---
6. At the break-even point:
Correct Answer: total revenue equals total cost
Rationale: The break-even point is defined as the quantity where total revenue
equals total cost, resulting in zero profit.
---
7. Belmont has bids from two companies to build its new indoor sports facility. It
will choose between Company A and Company B. Company A has a fixed cost of
75,000 and a variable cost of 15. Company B's fixed cost is 90,000 and variable
cost is 11. Use the total cost equation for each company and compare them. At
what volume is the total cost the same, that is at what volume is Belmont indifferent
between the two bidders?
Correct Answer: 3,750
, Rationale: Set Total Cost A = Total Cost B: 75,000 + 15Q = 90,000 + 11Q →
4Q = 15,000 → Q = 3,750.
---
8. Cully Furniture buys two products for resale: big shelves (B) and medium
shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage
space, and each medium shelf costs $300 and requires 90 cubic feet of storage
space. The company has $75,000 to invest in shelves this week, and the warehouse
has 18,000 cubic feet available for storage. Profit for each big shelf is $300 and
for each medium shelf is $150. What is the maximum profit?
Correct Answer: $45,000
Rationale: This requires solving the LP: Max Z = 300B + 150M subject to 500B +
300M ≤ 75,000 and 100B + 90M ≤ 18,000. The optimal solution is B = 150, M =
0, giving Z = $45,000.
---
9. Cully Furniture buys two products for resale: big shelves (B) and medium
shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage
space, and each medium shelf costs $300 and requires 90 cubic feet of storage
space. The company has $75,000 to invest in shelves this week, and the warehouse
has 18,000 cubic feet available for storage. Profit for each big shelf is $300 and
for each medium shelf is $150. In order to maximize profit, how many big shelves
(B) and how many medium shelves (M) should be purchased?
Correct Answer: B = 150, M = 0
