Worksheet: Using Systems of Equations to Investigate Profits

In this worksheet, we will practice using systems of equations to investigate the profits of a manufacturer.

Q1:

A stuffed animal business has a cost function of 𝐶 = 1 2 𝑥 + 3 0 and a revenue function 𝑅 = 2 0 𝑥 . First, find the break-even point. Then, calculate the corresponding revenue.

  • A 𝑥 = 0 , 𝑅 = 7 5
  • B 𝑥 = 7 5 , 𝑅 = 0
  • C 𝑥 = 7 5 , 𝑅 = 1 5 4
  • D 𝑥 = 1 5 4 , 𝑅 = 7 5

Q2:

A guitar factory has a cost of production given by 𝐶 ( 𝑥 ) = 7 5 𝑥 + 5 0 0 0 0 . If the company needs to break even after 150 units sold, at what price should they sell each guitar? Give your answer to the nearest dollar.

  • A $ 6 6 7
  • B $ 4 0 8
  • C $ 6 6 6
  • D $ 4 0 9

Q3:

A musician charges 𝐶 ( 𝑥 ) = $ ( 6 4 𝑥 + 2 0 0 0 0 ) , where 𝑥 is the total number of attendees at the concert. The venue charges $80 per ticket. After how many sold tickets does the venue break even, and what is the value of the total tickets sold at that point?

  • A 2 0 0 0 tickets, $ 1 6 0 0 0 0
  • B139 tickets, $ 1 1 1 2 0
  • C 1 5 0 0 tickets, $ 1 1 6 0 0 0
  • D 1 2 5 0 tickets, $ 1 0 0 0 0 0

Q4:

A cell phone factory produces 𝑥 cellphones with cost 𝐶 ( 𝑥 ) = 1 5 0 𝑥 + 1 0 0 0 0 and revenue 𝑅 ( 𝑥 ) = 2 0 0 𝑥 . What is the breakeven point?

  • A 𝑥 = 0
  • B 𝑥 = 2 0 0 7
  • C 𝑥 = 1 0 0 7
  • D 𝑥 = 2 0 0
  • E 𝑥 = 4 0 0 0 0

Q5:

A fast-food restaurant has a cost of production 𝐶 ( 𝑥 ) = 1 1 𝑥 + 1 2 0 and a revenue function 𝑅 ( 𝑥 ) = 5 𝑥 . When does the company start to turn a profit?

  • Awhen 𝑥 = 1 0 0
  • Bwhen 𝑥 = 2 0
  • Cwhen 𝑥 = 7 . 5
  • Dnever

Q6:

A laptop company has discovered that their cost and revenue functions for each day are 𝐶 ( 𝑥 ) = 3 𝑥 1 0 𝑥 + 2 0 0 2 and 𝑅 ( 𝑥 ) = 2 𝑥 + 1 0 0 𝑥 + 5 0 2 . Find the maximum and minimum number of laptops they could produce each day while still making a profit.

Hint: Your answers should be integers.

  • Aminimum: 2 laptops, maximum: 21 laptops
  • Bminimum: 1 laptop, maximum: 21 laptops
  • Cafter 51 computers
  • Dminimum: 2 laptops, maximum: 20 laptops

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