Maximizing Area with a 10m Wire: Square or Equilateral Triangle?

[Greywolfe1982]

Homework Statement

How do you cut a 10m piece of wire to get the maximum area when forming a square and an equilateral triangle?

Homework Equations

a_square=s²
a_triangle=s²*\sqrt{}3/4

The Attempt at a Solution

(Note that I've already found the total area function, its derivative, its critical points and that x=~8.73 is a minimum.) As far as I know, the maximum area comes from not cutting the wire at all and making a triangle. Using those area formulas, it gives an area of ~43.3m². However, the back of the book says maximum area comes from making a square. But doesn't this only give an area of 6.25m²? (s=10/4m, s²=6.25m²) Is the book wrong, or am I overlooking something?

 

[Mentallic]

Your formulae were correct, but you must've plugged the wrong value of s in when calculating the area of the triangle. Since it's an equilateral triangle, s=10/3 and plugging this into the triangle area formula A=\frac{s\sqrt{3}}{4} gives A\approx 4.8 which is less than the area of the square.

You might find this interesting to note, if you consider a circle as being a regular polygon with infinite sides, then for any given perimeter P, the more sides the polygon has, the larger its area. For example, an equilateral triangle with perimeter P (each side P/3) has less area than a square with sides P/4 each, which is less than a pentagon... etc. and the circle with perimeter P has the largest possible area.

 

[Greywolfe1982]

Aww. You beat me to it.

I was in the shower when I realized I used 10 as a side instead of 10/3. I haven't worked it out yet, but I'm assuming I won't have any problems from here. Thanks for the response.

 

[Mentallic]

Ahh that shower... we'd still be counting with our fingers if it weren't for that invention