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
riverxxy

A farmer has 320 m of fencing to enclose a rectangular field along a straight river. No fence is needed along the river. Find the largest area A that can be enclosed.

320 m2x + y = 320
A = m²
3.Medium
rh

A closed cylindrical can (with a top and a bottom) must hold 750 cm³. The metal used is proportional to the total surface area. Find the height h of the can that uses the least material.

V = 750 cm³
h = cm
4.Medium
xhxno lid

An open box (no lid) with a square base must hold 108000 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 = 108000 cm³
S = cm²
5.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) = 100 + 4q + 0.5q² euros. Find the output q that maximises profit.

p = €25C = 100 + 4q + 0.5q²
q = units
6.Hard
qP(q)maximum

A company sells q units per week. The price at which it can sell them is p = 90 − 0.2q euros per unit, and the cost is C(q) = 600 + 55q euros. Find the weekly output q that maximises profit.

p = 90 − 0.2qC = 600 + 55q
q = units

Answer key

  1. 1. V = 144 cm³
  2. 2. A = 12800 m²
  3. 3. h = 9.847 cm
  4. 4. S = 10800 cm²
  5. 5. q = 21 units
  6. 6. q = 87.5 units

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