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Worksheet: Optimization: Maximum Profit

Q1:

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 π‘₯ 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?

  • A0 chairs, 32 tables
  • B18 chairs, 38 tables
  • C38 chairs, 18 tables
  • D45 chairs, 0 tables
  • E32 chairs, 0 tables

Q2:

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 π‘₯ 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 ( π‘₯ , 𝑦 ) = 8 π‘₯ + 1 0 𝑦
  • B 𝑓 ( π‘₯ , 𝑦 ) = 2 . 5 π‘₯ + 3 . 5 𝑦 βˆ’ 2 4 0
  • C 𝑓 ( π‘₯ , 𝑦 ) = 2 π‘₯ + 1 . 5 𝑦
  • D 𝑓 ( π‘₯ , 𝑦 ) = 2 π‘₯ + 1 . 5 𝑦 + 2 4 0
  • E 𝑓 ( π‘₯ , 𝑦 ) = 2 π‘₯ + 1 0 𝑦 βˆ’ 4 8 0

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

  • A89 solid-color shirts and 69 tie-dye shirts
  • B0 solid-color shirts and 48 tie-dye shirts
  • C40 solid-color shirts and 40 tie-dye shirts
  • D48 solid-color shirts and 0 tie-dye shirts
  • E69 solid-color shirts and 40 tie-dye shirts