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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Name: ____________________ Date: ____________

1.Easy

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 largest possible volume V.

48 cm × 48 cm

V = cm³

2.Easy

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 y of the side parallel to the river for the largest area.

120 m2x + y = 120

y = m

3.Medium

An open box (no lid) with a square base must hold 108000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the base side x.

V = 108000 cm³

x = cm

4.Medium

An open box (no lid) with a square base must hold 13500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the height h.

V = 13500 cm³

h = cm

5.Hard

A company sells q units per week. The price at which it can sell them is p = 140 − 0.25q euros per unit, and the cost is C(q) = 5400 + 45q euros. Find the maximum weekly profit P.

p = 140 − 0.25qC = 5400 + 45q

P = €

6.Hard

A theatre sells 1050 tickets per show at €20 each. Market research shows that every €1 increase in the ticket price means 25 fewer tickets sold (and every €1 decrease, 25 more). Find the maximum revenue R per show.

1050 × €2025 / €1

R = €

Answer key

1. V = 8192 cm³
2. y = 60 m
3. x = 60 cm
4. h = 15 cm
5. P = 3625 €
6. R = 24025 €

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