Math CentralQuandaries & Queries


Question from Lyndsay, a teacher:

A rectangle is to be constructed having the greatest possible area and a perimeter of 50 cm.

(a) If one of the sides of the rectangle measures 'x' cm, find a formula for calculating the area of the rectangle as a function of 'x'.

(b) Determine the dimensions of the rectangle for which it has the greatest area possible. What is the maximum area?

Hi Lyndsay,

Suppose the other side has length $y$ cm then the perimeter is $x + x + y + y = 2x + 2y$ cm. But you know this is 50 cm so $2x + 2y = 50 \mbox{ or } x + y = 25.$ Thus $y = 25 - x.$

The area, $A$ of a rectangle is the length times the width and hence $A = x \times y$ or

\[A = x(25 - x).\]

There are a couple of ways to approach part (b). If you know some calculus you can treat part (b) as a max-min problem. Otherwise you can use the fact that the maximum or minimum of the quadratic function $a x^2 + b x + c$ is at the vertex.


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