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 80 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.

80 m2x + y = 80
A = m²
2.Easy
xxyx

A rectangular pen is fenced with 72 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.

72 m3x + 2y = 72
A = m²
3.Medium
rh

A closed cylindrical can (with a top and a bottom) must hold 500 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.

V = 500 cm³
r = cm
4.Medium
xhxno lid

An open box (no lid) with a square base must hold 32000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the height h.

V = 32000 cm³
h = 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) = 1100 + 10q + 0.5q² euros. Find the maximum profit P.

p = €70C = 1100 + 10q + 0.5q²
P = €
6.Hard
qP(q)maximum

A company sells q units per week. The price at which it can sell them is p = 180 − 1q euros per unit, and the cost is C(q) = 2700 + 15q euros. Find the price p that maximises profit.

p = 180 − 1qC = 2700 + 15q
p = €

Answer key

  1. 1. A = 800 m²
  2. 2. A = 216 m²
  3. 3. r = 4.301 cm
  4. 4. h = 20 cm
  5. 5. P = 700 €
  6. 6. p = 97.5 €

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