maximum rectangular area

Practise maximum rectangular area with an original VectorPaths question, worked explanation and related VCE Methods practice.

4 practice marks

Original practice question

A rectangle has perimeter 48 m. Let one side have length x m. Find the dimensions that maximise its area, and the maximum area.

Worked explanation

• Build the model

  The other side is $24-x$, so $A(x)=x(24-x)$ on $0<x<24$.

• Find the stationary point

  $A\prime(x)=24 - 2 x=0$ gives $x=12$.

• Verify the maximum

  $A\prime\prime(x)=-2<0$; the area approaches zero at either endpoint.

• Calculate the result

  The dimensions are $12$ m by $12$ m and the area is $144$ square metres.

Dimensions $12\text{ m}\times12\text{ m}$; maximum area $144\text{ m}^2$.

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