Calculus & Vectors (MCV4U) · Derivatives and Their Applications

Optimization word problems

Use derivatives to solve maximum-area and maximum-volume word problems with clean answers. Unlimited questions, five difficulty levels, and a full worked solution every time — free.

Practice this skill Curriculum: MCV4U-DC2.1

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A farmer will use 48 m of fencing to build a rectangular pen against the side of a barn (no fence needed along the barn). What length, in metres, should each side perpendicular to the barn be to maximize the pen's area?

Answer: 12

See one solved, step by step

A farmer will use 52 m of fencing to build a rectangular pen against the side of a barn (no fence needed along the barn). What length, in metres, should each side perpendicular to the barn be to maximize the pen's area?

📘 Worked solution
1Let x = each side perpendicular to the barn. Parallel side: 52 - 2x. Area: A(x) = x(52 - 2x)Only two short sides and one long side need fencing — the barn covers the fourth side.
2A(x) = 52x - 2x2, A'(x) = 52 - 4xExpand, then differentiate to locate the maximum.
3A'(x) = 0 => x = 13Set the derivative to 0 and solve; A''(x) = -4 < 0 confirms a maximum.
4Perpendicular side = 13 m, parallel side = 52 - 2(13) = 26 m, area = 338 m2Substitute x = 13 back in to get both dimensions and the maximum area.

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