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

To find out how many scarves Arianna can make with an area of 4 square feet, we can use the concept of inverse variation.

Step-by-step explanation:

To find out how many scarves Arianna can make with an area of 4 square feet, we can use the concept of inverse variation. We know that the number of scarves (n) is inversely proportional to the area of each scarf (a). This can be represented by the equation n = k/a, where k is a constant.

Using the given information that she can make 16 scarves with an area of 3 square feet, we can substitute the values into the equation to find the value of k. 16 = k/3, which gives us k = 48.

Now we can use this value of k to determine the number of scarves she can make with an area of 4 square feet. n = 48/4 = 12. Therefore, she can make 12 scarves with an area of 4 square feet.

Answer 2

`\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`