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
x30 cm48 cmx

An open box is made from a rectangular sheet of cardboard 30 cm × 48 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the largest possible volume V.

30 cm × 48 cm
V = cm³
2.Easy
x18 cm48 cmx

An open box is made from a rectangular sheet of cardboard 18 cm × 48 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the largest possible volume V.

18 cm × 48 cm
V = cm³
3.Medium
rh

A closed cylindrical can (with a top and a bottom) must hold 330 cm³. The metal used is proportional to the total surface area. Find the height h of the can that uses the least material.

V = 330 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 minimum surface area S.

V = 62500 cm³
S = cm²
5.Hard
qP(q)maximum

A small firm sells its product at a fixed market price of €65 per unit. Producing q units costs C(q) = 300 + 46q + 0.1q² euros. Find the maximum profit P.

p = €65C = 300 + 46q + 0.1q²
P = €
6.Hard
qP(q)maximum

A small firm sells its product at a fixed market price of €35 per unit. Producing q units costs C(q) = 200 + 14q + 0.2q² euros. Find the maximum profit P.

p = €35C = 200 + 14q + 0.2q²
P = €

Answer key

  1. 1. V = 3888 cm³
  2. 2. V = 1600 cm³
  3. 3. h = 7.49 cm
  4. 4. S = 7500 cm²
  5. 5. P = 602.5 €
  6. 6. P = 351.3 €

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