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 40 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.

40 m2x + y = 40
A = m²
2.Easy
x48 cm48 cmx

An open box is made from a square sheet of cardboard 48 cm × 48 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.

48 cm × 48 cm
x = cm
3.Medium
rh

A closed cylindrical can (with a top and a bottom) must hold 400 cm³. The metal used is proportional to the total surface area. Find the smallest possible surface area A.

V = 400 cm³
A = cm²
4.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
5.Hard
qP(q)maximum

A company sells q units per week. The price at which it can sell them is p = 140 − 1q euros per unit, and the cost is C(q) = 1500 + 30q euros. Find the maximum weekly profit P.

p = 140 − 1qC = 1500 + 30q
P = €
6.Hard
qP(q)maximum

A company sells q units per week. The price at which it can sell them is p = 170 − 1q euros per unit, and the cost is C(q) = 700 + 50q euros. Find the price p that maximises profit.

p = 170 − 1qC = 700 + 50q
p = €

Answer key

  1. 1. A = 200 m²
  2. 2. x = 8 cm
  3. 3. A = 300.5 cm²
  4. 4. A = 219.7 cm²
  5. 5. P = 1525 €
  6. 6. p = 110 €

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