1.Easy
An open box is made from a rectangular sheet of cardboard 21 cm × 45 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the largest possible volume V.
21 cm × 45 cm
V = cm³
Optimization worksheet with answers
2.Easy
An open box is made from a square sheet of cardboard 36 cm × 36 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.
36 cm × 36 cm
x = cm
3.Medium
An open box (no lid) with a square base must hold 32000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the height h.
V = 32000 cm³
h = cm
4.Medium
An open box (no lid) with a square base must hold 62500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the minimum surface area S.
V = 62500 cm³
S = cm²
5.Hard
A small firm sells its product at a fixed market price of €25 per unit. Producing q units costs C(q) = 100 + 4q + 0.5q² euros. Find the maximum profit P.
p = €25C = 100 + 4q + 0.5q²
P = €
6.Hard
A small firm sells its product at a fixed market price of €30 per unit. Producing q units costs C(q) = 5000 + 10q + 0.01q² euros. Find the output q that maximises profit.
p = €30C = 5000 + 10q + 0.01q²
q = units
Answer key
- 1. V = 1944 cm³
- 2. x = 6 cm
- 3. h = 20 cm
- 4. S = 7500 cm²
- 5. P = 120.5 €
- 6. q = 1000 units
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