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2006 Paper 1 Q2
D: 1516.0 B: 1500.0
optimisation area circular motion tethered goat problem inequality geometric proof arc length and sector area

A small goat is tethered by a rope to a point at ground level on a side of a square barn which stands in a large horizontal field of grass. The sides of the barn are of length \(2a\) and the rope is of length \(4a\). Let \(A\) be the area of the grass that the goat can graze. Prove that \(A\le14\pi a^2\) and determine the minimum value of \(A\).

Solution:

TikZ diagram
The areas are \(8\pi a^2 + \frac14 \pi (4a-x)^2 + \frac14 \pi (2a-x)^2 + \frac14\pi(2a+x)^2+\frac14 \pi x^2\) ie \begin{align*} A &= \frac{\pi}{4} \left ( x^2 \left (1 + 1 + 1 + 1 \right) + x \left (4a-4a-8a \right)+\left (32a^2+16a^2+4a^2+4a^2 \right)\right) \\ &= \frac{\pi}{4} \left (4x^2-8ax+56a^2 \right) \\ &= \pi(x^2-2ax+14a^2) \\ &= \pi ((x-a)^2+13a^2) \end{align*} Since \(x \in [0, 2a]\) we have \(13\pi a^2 \leq A \leq 14 \pi a^2\)

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