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

A farmer has 200 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.

200 m2x + y = 200

x = m

2.Easy

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

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 smallest possible surface area A.

V = 500 cm³

A = cm²

4.Medium

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 height h of the can that uses the least material.

V = 500 cm³

h = cm

5.Hard

A small firm sells its product at a fixed market price of €25 per unit. Producing q units costs C(q) = 100 + 16q + 0.1q² euros. Find the maximum profit P.

p = €25C = 100 + 16q + 0.1q²

P = €

6.Hard

A theatre sells 1700 tickets per show at €10 each. Market research shows that every €1 increase in the ticket price means 50 fewer tickets sold (and every €1 decrease, 50 more). Find the ticket price p that maximises revenue.

1700 × €1050 / €1

p = €

Answer key

1. x = 50 m
2. A = 216 m²
3. A = 348.7 cm²
4. h = 8.603 cm
5. P = 102.5 €
6. p = 22 €

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