# Two shapes and one line.

• Sep 30th 2012, 05:34 PM
Nervous
Two shapes and one line.
Quote:

A piece of wire of length 60 inches is cut into two pieces. One piece is foldedinto a square and the other into an equilateral triangle. Find the maximum of the area of the square
plus the area of the triangle. Choose the closest number from the list below.
The furthest I've gotten is

$60 = 4x + 3y$

PS, this is just for a test I'm taking online, I'm not even in a calculus class. Don't think I'm trying to cheat or anything.
• Sep 30th 2012, 05:47 PM
Prove It
Re: Two shapes and one line.
Quote:

Originally Posted by Nervous
The furthest I've gotten is

$60 = 4x + 3y$

PS, this is just for a test I'm taking online, I'm not even in a calculus class. Don't think I'm trying to cheat or anything.

Well if you cut the wire at length x, then the remaining length is 60 - x.

If you make the square with the length 60 - x, then its side length is \displaystyle \begin{align*} \frac{60 - x}{4} \end{align*} and its area is \displaystyle \begin{align*} \left(\frac{60-x}{4}\right)^2 = \frac{3600 - 120x + x^2}{16} \end{align*}

If the triangle is made with the length x, then its base is x and its height is \displaystyle \begin{align*} \frac{\sqrt{3}\,x}{2} \end{align*}, therefore its area is \displaystyle \begin{align*} \frac{\sqrt{3}\,x^2}{4} \end{align*}

So the total area is \displaystyle \begin{align*} \frac{3600 - 120 + \left(4\sqrt{3} + 1\right)x^2}{16} \end{align*}

Can you figure out how to find the maximum area now?
• Oct 1st 2012, 05:40 PM
Nervous
Re: Two shapes and one line.
I derived the fraction and got:

$\frac{-120+2x+8\sqrt{3}x^2}{16}$

I set it equal to zero and got:

$x=~7.568$

And when I plugged that into the original equation, I got ~196.620

But, the answer is supposed to be 225...
• Oct 1st 2012, 05:57 PM
MaxJasper
Re: Two shapes and one line.
You equated area to zero! You should equate its derivative to zero in order to obtain x value that makes area maximum $\frac{60}{1+4 \sqrt{3}}$ or something like that!
• Oct 2nd 2012, 03:57 AM
Nervous
Re: Two shapes and one line.
$\frac{60}{1+4 \sqrt{3}} = ~7.568$