Question

A company is required to fence off the area around a robot arm to comply with health and safety law. They have 750m of fencing available. Find the maximum area they can fence off?

Answer 1

We can see that the maximum perimeter of the area is 750 m.

First, let us consider the rectangular shape with the length $l$ and width $w$ . Therefore,

$p=2(l+w), \; 750 = 2(l+w), \; l+w = 375.$ (1)

We should find maximum of area, i.e. maximum of $S(l,w)=l\cdot w.$ From (1) we write $w=375-l$ , so area is $S(l) = l(375-l) = -l^2 + 375l.$

Let us maximize this function. We may take the derivative and write

$S'(l) = -2l+375.$

$S'(l) = 0$ when $l=0.5\cdot375 = 187.5\, m .$

Therefore, $w=375-l = 375-187.5 = 187.5\,m,$ so the area is a square and the maximum $S$ is $187.5^2 = 35156.25\, m^2.$

However, according to the Isoperimetric inequality (see https://en.wikipedia.org/wiki/Isoperimetric_inequality#On_a_plane) we can see that circle will be the figure with maximum area.

The perimeter of the circle of radius $R$ is $2\pi R$, so we can calculate the value of radius:

$R = \dfrac{750}{2\pi} \approx 119.37\,m.$

The area will be

$S(R) = \pi R^2 = \pi \cdot 119.37^2 = 44762.33\,m^2.$

We may note that circle has significantly larger area.