Maximize area with a quadratic model
Use the vertex of a quadratic area model to find the maximum area of a fenced rectangular pen. Unlimited questions, five difficulty levels, and a full worked solution every time — free.
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A farmer has 40 m of fencing to build a rectangular pen against the side of a barn, so only three sides need fencing. What is the maximum possible area of the pen, in square metres?
A(w) = w(40 - 2w)
Answer: 200
See one solved, step by step
A farmer has 20 m of fencing to build a rectangular pen against the side of a barn, so only three sides need fencing. What is the maximum possible area of the pen, in square metres?
A(w) = w(20 - 2w)
📘 Worked solution
1Let w = width. The barn covers one length side, so fencing covers 2 widths + 1 length: length = 20 - 2w.Only 3 sides need fencing since the barn wall forms the fourth side.
2A(w) = w(20 - 2w) = −2w2 + 20wArea = width * length; expand it into standard quadratic form.
3Vertex at w = −b/(2a) = −20 / (2 · (−2)) = 5The vertex of a downward-opening parabola gives the maximum area.
4Length = 20 - 2(5) = 10 m, Area = 5 · 10 = 50 m2Substitute the best width back in to find the maximum area.
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