2026 ZHSS P2 Q5

A farmer wants to enclose a rectangular pen against a long, straight barn wall. The barn wall forms the entire top side of the pen, so no fencing is needed there.

Fencing is required for:

• the bottom longer side (length \(L\) metres), using standard fencing at $1 per metre.
• the two shorter sides and one internal dividing fence (each of length \(x\) metres), using reinforced fencing at $4 per metre.

The internal dividing fence runs parallel to the shorter sides and splits the pen into two equal sections.

The farmer has a total budget of $120.

The diagram below shows the rectangular pen.

1. Show that the total area of the pen, \(A\text{ m}^2\), is given by \(A=120x-12x^2\).[3]
2. Hence, find the maximum area of the pen. Justify that the area found is a maximum.[4]