1.Easy
An open box is made from a rectangular sheet of cardboard 30 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.
30 cm × 48 cm
V = cm³
Optimization worksheet with answers
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
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
4.Medium
A closed cylindrical can (with a top and a bottom) must hold 1000 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.
V = 1000 cm³
r = cm
5.Hard
A company sells q units per week. The price at which it can sell them is p = 110 − 0.25q euros per unit, and the cost is C(q) = 1000 + 70q euros. Find the price p that maximises profit.
p = 110 − 0.25qC = 1000 + 70q
p = €
6.Hard
A small firm sells its product at a fixed market price of €70 per unit. Producing q units costs C(q) = 700 + 12q + 0.25q² euros. Find the output q that maximises profit.
p = €70C = 700 + 12q + 0.25q²
q = units
Answer key
- 1. V = 3888 cm³
- 2. A = 216 m²
- 3. S = 7500 cm²
- 4. r = 5.419 cm
- 5. p = 90 €
- 6. q = 116 units
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