Optimization worksheet with answers

Free printable Optimization worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
x10 cm16 cmx

An open box is made from a rectangular sheet of cardboard 10 cm × 16 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the largest possible volume V.

10 cm × 16 cm
V = cm³
2.Easy
x30 cm30 cmx

An open box is made from a square sheet of cardboard 30 cm × 30 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the cut x that gives the largest volume.

30 cm × 30 cm
x = cm
3.Medium
xhxno lid

An open box (no lid) with a square base must hold 4000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the height h.

V = 4000 cm³
h = cm
4.Medium
xhxno lid

An open box (no lid) with a square base must hold 62500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the base side x.

V = 62500 cm³
x = cm
5.Hard
qP(q)maximum

A small firm sells its product at a fixed market price of €70 per unit. Producing q units costs C(q) = 3200 + 30q + 0.05q² euros. Find the output q that maximises profit.

p = €70C = 3200 + 30q + 0.05q²
q = units
6.Hard
qP(q)maximum

A small firm sells its product at a fixed market price of €25 per unit. Producing q units costs C(q) = 1700 + 10q + 0.02q² euros. Find the output q that maximises profit.

p = €25C = 1700 + 10q + 0.02q²
q = units

Answer key

  1. 1. V = 144 cm³
  2. 2. x = 5 cm
  3. 3. h = 10 cm
  4. 4. x = 50 cm
  5. 5. q = 400 units
  6. 6. q = 375 units

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