Problem Statement
A farmer wishes to enclose a rectangular vegetable garden using $160\text{ m}$ of fencing against an existing straight stone wall. The wall forms one length of the rectangle, so fencing is only required on three sides.
**(a)** [2 marks] Let $x$ be the width of the garden perpendicular to the wall. Express the area $A(x)$ in terms of $x$.
**(b)** [4 marks] Find the value of $x$ that maximizes the garden area and state the maximum area.
Verified Solution & Marking Scheme
Part (a) [M1]
Perimeter: $2x + \text{length} = 160 \implies \text{length} = 160 - 2x$
Part (a) [A1]
$A(x) = x(160 - 2x) = 160x - 2x^2$
Part (b) [M1]
Setting $A'(x) = 160 - 4x = 0$ or finding vertex $x = -b / (2a)$
Part (b) [A1]
$x = 40\text{ m}$
Part (b) [R1]
$A''(x) = -4 < 0$, verifying local maximum
Part (b) [A1]
$A_{\max} = 40(160 - 80) = 3200\text{ m}^2$