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
x9 cm24 cmx

An open box is made from a rectangular sheet of cardboard 9 cm × 24 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the largest possible volume V.

9 cm × 24 cm
V = cm³
2.Easy
x9 cm24 cmx

An open box is made from a rectangular sheet of cardboard 9 cm × 24 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.

9 cm × 24 cm
x = cm
3.Medium
xhxno lid

An open box (no lid) with a square base must hold 13500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the height h.

V = 13500 cm³
h = cm
4.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 radius r that uses the least material.

V = 330 cm³
r = cm
5.Hard
qP(q)maximum

A company sells q units per week. The price at which it can sell them is p = 200 − 2q euros per unit, and the cost is C(q) = 700 + 60q euros. Find the weekly output q that maximises profit.

p = 200 − 2qC = 700 + 60q
q = units
6.Hard
qP(q)maximum

A company sells q units per week. The price at which it can sell them is p = 70 − 0.1q euros per unit, and the cost is C(q) = 1300 + 20q euros. Find the maximum weekly profit P.

p = 70 − 0.1qC = 1300 + 20q
P = €

Answer key

  1. 1. V = 200 cm³
  2. 2. x = 2 cm
  3. 3. h = 15 cm
  4. 4. r = 3.745 cm
  5. 5. q = 35 units
  6. 6. P = 4950 €

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