Rectangular garden enclosure

00:00 6 marks

A rectangular garden is to be enclosed by a fence. One side of the garden is against an existing wall, so fencing is only needed for the other three sides. The total length of fencing available is 40 metres. Let the width of the garden (perpendicular to the wall) be \(x\) metres.

(a) Show that the area, \(A\) m², of the garden is given by \(A = 40x – 2x^2\). [2]

(b) Find the dimensions of the garden that maximize its area. [4]

Part a

3 minutes 2 marks

Question

Show that the area, \(A\) m², of the garden is given by \(A = 40x – 2x^2\). [2]

Hint Hint

Hint

Express the third side in terms of \(x\) using the fencing constraint, then write \(A = x\times y\).

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation

Answer Answer

Complete Answer

Using \(2x + y = 40\) ⇒ \(y = 40 – 2x\). Hence \(A = x(40 – 2x) = 40x – 2x^2\).

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Part b

5 minutes 4 marks

Question

Find the dimensions of the garden that maximize its area. [4]

Hint Hint

Hint

Differentiate \(A = 40x – 2x^2\) and set \(dA/dx = 0\), then find \(y\).

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation

Answer Answer

Complete Answer

Differentiating gives \(dA/dx = 40 – 4x\). Setting to zero yields \(x = 10\) m; then \(y = 40 – 2(10) = 20\) m. Dimensions: 10 m by 20 m.

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation