Question

Helpp! -- A piece of wire 5 inches long is to be cut in two pieces. One piece is x inches long and is to be bent into the shape of a square. The other piece is to be bent in the shape of a circle. Find an expression for the total area made up by the square and the circle as a function of x.

Answer 1

Check the picture below.

since, we know that for the circle, the perimeter is s+s+s+s or 4s, then 4s = x and so on.

now, for the circle, the perimeter is just the circumference, circumference of a circle is 2πr, thus 2πr = 5-x and so on.

anyway... so... the equation of the area made by those two in x-terms, is just their sum... so let's do that then

$\bf \pi \left( \cfrac{5-x}{2\pi } \right)^2\quad +\quad \cfrac{x^2}{4^2}\\\\ -------------------------------\\\\ \pi\cdot \cfrac{(5-x)^2}{(2\pi )^2}+\cfrac{x^2}{4^2}\implies \cfrac{\pi (5-x)^2}{2^2\pi^2}+\cfrac{x^2}{4^2}\implies \cfrac{(5-x)^2}{4\pi}+\cfrac{x^2}{4^2} \\\\\\ \textit{let's use the LCD of }4^2\pi \textit{ to add them up} \\\\\\ \cfrac{4(5-x)^2+\pi x^2}{4^2\pi }\implies \cfrac{4(5-x)^2+\pi x^2}{16\pi }$

now, you can expand the squared binomial on the numerator, but there will not be any like-terms to simplify, so, it won't make much difference, so.... you can expand it or not.