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 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
2.Easy
xxyx

A rectangular pen is fenced with 360 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 length x that gives the largest total area.

360 m3x + 2y = 360
x = m
3.Medium
rh

A closed cylindrical can (with a top and a bottom) must hold 250 cm³. The metal used is proportional to the total surface area. Find the smallest possible surface area A.

V = 250 cm³
A = cm²
4.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 radius r that uses the least material.

V = 750 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 = 80 − 2q euros per unit, and the cost is C(q) = 200 + 20q euros. Find the price p that maximises profit.

p = 80 − 2qC = 200 + 20q
p = €
6.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) = 3600 + 8q + 0.01q² euros. Find the maximum profit P.

p = €25C = 3600 + 8q + 0.01q²
P = €

Answer key

  1. 1. A = 12800 m²
  2. 2. x = 60 m
  3. 3. A = 219.7 cm²
  4. 4. r = 4.924 cm
  5. 5. p = 50 €
  6. 6. P = 3625 €

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