Chapter 5, Problem 23CT

Solution

To calculate : The pair of shoes that can fit in the closet floor whose area is $18$ square feet.

The number of pairs of shoes that can fit if the area of the closet floor is $18$ square feet is approximately $39$ .

Given information :

The number of shoes bought by E varies directly as the square of the area of the floor of her closet. $12$ pair of shoes can fit when the closet was $10$ square feet.

Formula used:

If $xandy$ are directly proportionate then,

  $x\propto y$

  $\begin{array}{l}x=ky,k\ne 0\\ \frac{x}{y}=k\end{array}$

where, $k$ is the proportionality constant. If $k>0$ , then $xandy$ both increase simultaneously and if $k<0$ , then $xandy$ both decrease simultaneously.

Calculation :

It is given that the area of the closet floor is $10$ square feet if she buys $12$ pair of shoes.

So, find the value of the proportionality constant i.e

  $\begin{array}{l}k=\frac{x}{y^2}\\ =\frac{12}{{10}^2}\left\{\text{Given }x=12\text{ and }y=10\right\}\\ =\frac{3}{25}\left\{\text{Solve further}\right\}\end{array}$

Now,

If the area of the closet floor is $18$ square feet, find the number of pairs of shoes that Emily can fit in.

Calculate the value of $x$ when $y=18$ using proportionality constant.

  $\begin{array}{l}\frac{x}{y^2}=k\left\{\text{Proportionality constant}\right\}\\ \Rightarrow \frac{x}{{18}^2}=\frac{3}{25}\left\{\text{From above}\right\}\\ \Rightarrow \frac{x}{324}=\frac{3}{25}\left\{\because {18}^2=324\right\}\\ \Rightarrow x=\frac{3\times 324}{25}\left\{\text{Use the cross product rule}\right\}\\ \Rightarrow x=38.88\\ \therefore x=39\left\{\text{approximately}\right\}\end{array}$

Therefore, the number of pairs of shoes that can fit if the area of the closet floor is $18$ square feet is approximately $39$ .