Fencing optimisation with a constraint

Practise fencing optimisation with a constraint with an original VectorPaths question, worked explanation and related VCE Methods practice.

5 practice marks

Original practice question

A rectangular garden is built against a straight wall. The other three sides use 40 m of fencing. Let x be the length of each side perpendicular to the wall. (a) Write the area A as a function of x and state its physical domain. [2 marks] (b) Find the dimensions giving maximum area and justify the maximum. [3 marks]

Worked explanation

• Part (a)

  The remaining side is $40-2x$ m. Both side lengths must be positive, so $0<x<20$. Multiply the lengths for area.

• Part (b)

  $A\prime=40-4x=0$ gives $x=10$. $A\prime\prime=-4<0$ and the endpoint areas tend to zero. The other side is 20, so maximum area is 200.

(a) $A(x)=x(40-2x)$, $0<x<20$.; (b) $10$ m by $20$ m; area $200$ square metres.

Original VectorPaths practice; not an official VCAA question or marking rubric.