Finding the maximum area of a shape.

Question

You have $3.0$ m of edging to put around a flower bed. Find the maximum area you can enclose if the flower bed is an equilateral triangle.

A) Find the maximum area you can enclose if the flower bed is a square.

B) Find the maximum area you can enclose if the flower bed is circular.

C) What property of circles makes them a useful shape for the base of storage tanks and some types of buildings?

Answer 1

Equilateral triangle:

Side length is $3/3=1$m, area of equilateral triangle is given by $\dfrac{s^2\sqrt3}4$ so area is $\boxed{\dfrac{\sqrt3}4\approx0.433}$

Square:

Side length is $3/4=\frac 34$m, area of square is given by $s^2$ so area is $\boxed{\dfrac 9{16}\approx0.563}$

Circle:

Circumference of $3$ gives radius of $\dfrac3{2\pi}$, area of circle is given by $\pi r^2$, so area is $\boxed{\dfrac9{4\pi}\approx0.716}$

Storage containment structures are round due to the fact that round structures have the ability to withstand more stress. Also, as you can see from the results, they can maximize the area of a structure using the least amount of material.