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 24 cm × 24 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the largest possible volume V.

24 cm × 24 cm

V = cm³

2.Easy

An open box is made from a square sheet of cardboard 42 cm × 42 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the largest possible volume V.

42 cm × 42 cm

V = cm³

3.Medium

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 height h of the can that uses the least material.

V = 250 cm³

h = cm

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 smallest possible surface area A.

V = 500 cm³

A = cm²

5.Hard

A small firm sells its product at a fixed market price of €50 per unit. Producing q units costs C(q) = 14500 + 16q + 0.01q² euros. Find the output q that maximises profit.

p = €50C = 14500 + 16q + 0.01q²

q = units

6.Hard

A company sells q units per week. The price at which it can sell them is p = 180 − 0.25q euros per unit, and the cost is C(q) = 300 + 150q euros. Find the maximum weekly profit P.

p = 180 − 0.25qC = 300 + 150q

P = €

Answer key

1. V = 1024 cm³
2. V = 5488 cm³
3. h = 6.828 cm
4. A = 348.7 cm²
5. q = 1700 units
6. P = 600 €

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