1.Easy
A farmer has 400 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.
400 m2x + y = 400
y = m
Optimization worksheet with answers
2.Easy
A rectangular pen is fenced with 240 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 length x that gives the largest total area.
240 m3x + 2y = 240
x = m
3.Medium
An open box (no lid) with a square base must hold 4000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the base side x.
V = 4000 cm³
x = cm
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 smallest possible surface area A.
V = 750 cm³
A = cm²
5.Hard
A theatre sells 490 tickets per show at €20 each. Market research shows that every €1 increase in the ticket price means 10 fewer tickets sold (and every €1 decrease, 10 more). Find the maximum revenue R per show.
490 × €2010 / €1
R = €
6.Hard
A company sells q units per week. The price at which it can sell them is p = 120 − 0.25q euros per unit, and the cost is C(q) = 500 + 90q euros. Find the weekly output q that maximises profit.
p = 120 − 0.25qC = 500 + 90q
q = units
Answer key
- 1. y = 200 m
- 2. x = 40 m
- 3. x = 20 cm
- 4. A = 457 cm²
- 5. R = 11903 €
- 6. q = 60 units
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