1.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
Optimization worksheet with answers
2.Easy
A rectangular pen is fenced with 360 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.
360 m3x + 2y = 360
A = m²
3.Medium
A closed cylindrical can (with a top and a bottom) must hold 400 cm³. The metal used is proportional to the total surface area. Find the smallest possible surface area A.
V = 400 cm³
A = cm²
4.Medium
A closed cylindrical can (with a top and a bottom) must hold 750 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.
V = 750 cm³
r = cm
5.Hard
A small firm sells its product at a fixed market price of €75 per unit. Producing q units costs C(q) = 52200 + 16q + 0.01q² euros. Find the output q that maximises profit.
p = €75C = 52200 + 16q + 0.01q²
q = units
6.Hard
A theatre sells 780 tickets per show at €20 each. Market research shows that every €1 increase in the ticket price means 20 fewer tickets sold (and every €1 decrease, 20 more). Find the ticket price p that maximises revenue.
780 × €2020 / €1
p = €
Answer key
- 1. y = 60 m
- 2. A = 5400 m²
- 3. A = 300.5 cm²
- 4. r = 4.924 cm
- 5. q = 2950 units
- 6. p = 29.5 €
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