4400 Higher June 2005 Paper 4 Q14

EdexcelHigherOld spec8 marksDifferentiationForming Equations

14 A farmer wants to make a rectangular pen for keeping sheep.
He uses a wall, \(AB\), for one side.
For the other three sides, he uses 28 m of fencing.
He wants to make the area of the pen as large as possible.

Rectangular pen against wall AB; the two sides perpendicular to the wall are x and the side parallel to the wall is 28 − 2x

Diagram NOT accurately drawn

The width of the pen is \(x\) metres.
The length parallel to the wall is \((28 - 2x)\) metres.

(a) The area of the pen is \(y\) m².
Show that \(y = 28x - 2x^2\). (1)
(b) For \(y = 28x - 2x^2\),
(i) find \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\),
(ii) find the value of \(x\) for which \(y\) is a maximum.
(iii) Explain how you know that this value gives a maximum. (5)
(c) Find the largest possible area of the pen. (2)
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