Question

arianna has a large piece of fabric that she wants to use to make some scarves. the number n of scarves she can make is inversely proportional to the area a of each scarf. if the area of each scarf is 3 square feet, then she can make 16 scarves. if the area of each scarf is 4 square feet, how many scarves can she make?

Answer 1

$\bf \qquad \qquad \textit{inverse proportional variation} \\\\ \textit{\underline{y} varies inversely with \underline{x}}\qquad \qquad y=\cfrac{k}{x}\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf \stackrel{\textit{\underline{n} scarves is inversely proportional to the area \underline{a} of each scarf}}{n=\cfrac{k}{a}} \\\\\\ \textit{we also know that } \begin{cases} a=3\\ n=16 \end{cases}\implies 16=\cfrac{k}{3}\implies 48=k~\hfill \boxed{n=\cfrac{48}{a}} \\\\\\ \textit{when a = 4, what is \underline{n}?}\qquad n=\cfrac{48}{4}\implies n=12$