Further Mathematics · AQA 7367 · Option D · Topic DD

Linear programming

Writing real problems as linear programs and solving them graphically and with the Simplex algorithm.

  • 4 sub-topics
  • 2 in Year 12, 2 in Year 13
  • 51 practice questions
  • 4 animated lessons
Practice questions (51) Lessons Worksheets All AQA Further Mathematics topics

What the specification says

Every sub-topic AQA can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

  1. DD1Formulate constrained optimisation problemsY12
  2. DD2Graphical solutionY12
  3. DD3Simplex algorithm with slack variablesY13
  4. DD4Interpret a simplex tableauY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 51 questions in this topic, free, no account needed.

Linear programming

Easy

Writing a constraint · Year 12 · 1 mark

  1. A workshop makes xx chairs and yy tables. Each chair takes 6 hours and each table takes 9 hours, and 180 hours are available. Which constraint models this?
    (1)

Show worked solutionHide worked solution
  1. The hours used cannot be more than the hours available.DD: 6x+9y≤1806x + 9y \leq 180

Total1

  • M
  • A
  • B
How did it go?

Report a problem with this question

Linear programming

Medium

Formulating a linear programming problem · Year 12 · 5 marks

A workshop makes chairs and tables. A chair takes 2 hours of cutting and 1 hour of finishing; a table takes 3 hours of cutting and 2 hours of finishing. Each week there are 120 hours of cutting time and 70 hours of finishing time. At least 2 times as many chairs as tables must be made. The profit is £12 on a chair and £15 on a table.
  1. Formulate this as a linear programming problem, defining your variables.
    (5)
Show worked solutionHide worked solution
  1. Let xx be the number of chairs and yy the number of tables made each week.
  2. maximise P=12x+15y\text{maximise } P = 12x + 15y
  3. One constraint for each resource.2x2x+3y≤120,  x{}+ 3y \leq 120, \ \ x+2y≤70{}+ 2y \leq 70
  4. x≥2y,  x,y≥0x \geq 2y, \ \ x, y \geq 0

Total5

  • M
  • A
  • B
How did it go?

Report a problem with this question

Linear programming

Hard

Graphical solution of a linear programming problem · Year 12 · 5 marks

Figure 1 shows the feasible region RR for the constraints x+y≤10x + y \leq 10, 3x+y≤213x + y \leq 21, x,y≥0x, y \geq 0.
xyR
Figure 1
  1. (a)
    Find the coordinates of the vertices of RR.
    (3)
  2. (b)
    Find the maximum value of P=2x+5yP = 2x + 5y on RR, and where it occurs.
    (2)
Show worked solutionHide worked solution

Part (a)3 marks

  1. Solve the pairs of boundary equations simultaneously.(0,0), (0,10), (112,92), (7,0)(0, 0), \ (0, 10), \ (\frac{11}{2}, \frac{9}{2}), \ (7, 0)

Part (b)2 marks

  1. Test each vertex, or slide an objective line.P(0,0)P(0, 0)=0, P(0,10){}= 0, \ P(0, 10)=50, P(112,92){}= 50, \ P(\frac{11}{2}, \frac{9}{2})=672, P(7,0){}= \frac{67}{2}, \ P(7, 0)=14{}= 14
  2. Pmax⁡=50 at (0,10)P_{\max} = 50 \text{ at } (0, 10)

Total5

  • M
  • A
  • B
How did it go?

Report a problem with this question