1.Easy
A farmer has 240 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.
240 m2x + y = 240
y = m
Optimization worksheet with answers
2.Easy
A farmer has 240 m of fencing to enclose a rectangular field along a straight river. No fence is needed along the river. Find the largest area A that can be enclosed.
240 m2x + y = 240
A = m²
3.Medium
An open box (no lid) with a square base must hold 108000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the height h.
V = 108000 cm³
h = cm
4.Medium
A rectangular plot of 1200 m² is fenced on all four sides. The side along the street, of length x, costs €30 per metre; the other three sides cost €10 per metre. Find the length x of the street side that minimises the cost.
A = 1200 m²€30/m€10/m
x = m
5.Hard
A small firm sells its product at a fixed market price of €80 per unit. Producing q units costs C(q) = 300 + 68q + 0.05q² euros. Find the output q that maximises profit.
p = €80C = 300 + 68q + 0.05q²
q = units
6.Hard
A small firm sells its product at a fixed market price of €70 per unit. Producing q units costs C(q) = 500 + 28q + 0.25q² euros. Find the output q that maximises profit.
p = €70C = 500 + 28q + 0.25q²
q = units
Answer key
- 1. y = 120 m
- 2. A = 7200 m²
- 3. h = 30 cm
- 4. x = 24.49 m
- 5. q = 120 units
- 6. q = 84 units
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