A piece of wire of length 50 m is cut into two parts. Each part is then bent to form a square. It is found that the total area of the square is 100 square meter. Find the difference in length of the sides of the two squares.
okay
when you cut the peice of wire let the peices be x and 50-x metes long (that works because 50-x+x=50)
so
one of the squares will have side lengths of x/4 metres and the other will have sides of (50-x)/4 metres
So
The areas if the squares will be $$\left(\frac{x}{4}\right)^2+\left(\frac{50-x}{4}\right)^2$$
so
$$\begin{array}{rll}
\left(\frac{x}{4}\right)^2+\left(\frac{50-x}{4}\right)^2&=&100\\\\
\frac{x^2}{16}+\frac{2500-100x+x^2}{16}&=&100\\\\
\frac{2x^2-100x+2500}{16}&=&100\\\\
2x^2-100x+2500&=&1600\\\\
2x^2-100x+900&=&0\\\\
x^2-50x+450&=&0\\\\
x&=&\frac{50\pm\sqrt{2500-1800}}{2}\\\\
x&=&\frac{50\pm\sqrt{700}}{2}\\\\
x&=&\frac{50\pm10\sqrt{7}}{2}\\\\
x&=&25\pm5\sqrt{7}\\\\
\end{array}$$
now one of the peices will be $$25+5\sqrt7$$ and the other will be $$25-5\sqrt7$$
the difference between these will be $$25+5\sqrt7-(25-5\sqrt7)=10\sqrt7\;m$$
the difference between the sides will be a quarter of this = $$2.5\sqrt7\;\;metres$$
okay
when you cut the peice of wire let the peices be x and 50-x metes long (that works because 50-x+x=50)
so
one of the squares will have side lengths of x/4 metres and the other will have sides of (50-x)/4 metres
So
The areas if the squares will be $$\left(\frac{x}{4}\right)^2+\left(\frac{50-x}{4}\right)^2$$
so
$$\begin{array}{rll}
\left(\frac{x}{4}\right)^2+\left(\frac{50-x}{4}\right)^2&=&100\\\\
\frac{x^2}{16}+\frac{2500-100x+x^2}{16}&=&100\\\\
\frac{2x^2-100x+2500}{16}&=&100\\\\
2x^2-100x+2500&=&1600\\\\
2x^2-100x+900&=&0\\\\
x^2-50x+450&=&0\\\\
x&=&\frac{50\pm\sqrt{2500-1800}}{2}\\\\
x&=&\frac{50\pm\sqrt{700}}{2}\\\\
x&=&\frac{50\pm10\sqrt{7}}{2}\\\\
x&=&25\pm5\sqrt{7}\\\\
\end{array}$$
now one of the peices will be $$25+5\sqrt7$$ and the other will be $$25-5\sqrt7$$
the difference between these will be $$25+5\sqrt7-(25-5\sqrt7)=10\sqrt7\;m$$
the difference between the sides will be a quarter of this = $$2.5\sqrt7\;\;metres$$