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
riverxxy

A farmer has 120 m of fencing to enclose a rectangular field along a straight river. No fence is needed along the river. Find the length x of each side perpendicular to the river for the largest area.

120 m2x + y = 120
x = m
2.Easy
xxyx

A rectangular pen is fenced with 144 m of fencing and divided into two parts by one extra fence parallel to the sides of length x (three fences of length x and two of length y in total). Find the largest total area A that can be enclosed.

144 m3x + 2y = 144
A = m²
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 500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the height h.

V = 500 cm³
h = cm
5.Hard
qP(q)maximum

A small firm sells its product at a fixed market price of €20 per unit. Producing q units costs C(q) = 1000 + 4q + 0.02q² euros. Find the output q that maximises profit.

p = €20C = 1000 + 4q + 0.02q²
q = units
6.Hard
qP(q)maximum

A company sells q units per week. The price at which it can sell them is p = 60 − 0.25q euros per unit, and the cost is C(q) = 500 + 30q euros. Find the maximum weekly profit P.

p = 60 − 0.25qC = 500 + 30q
P = €

Answer key

  1. 1. x = 30 m
  2. 2. A = 864 m²
  3. 3. h = 7.49 cm
  4. 4. h = 5 cm
  5. 5. q = 400 units
  6. 6. P = 400 €

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