Chapter 1.1, Problem 46E

a.

To determine

To write: and solve equation to find area of one rectangular mat.

Answer to Problem 46E

  ${\text{18 ft}}^2$

Explanation of Solution

Given:

The big square mat has area ${\text{81 ft}}^2$ .

This mat uses 4 identical rectangular mats and one small square mat.

The area of square mat is half the area of the rectangular mats.

Let the area of the rectangular mats be x square feet’s.

Then, the area of the small square mat is $\frac{x}{2}{\text{ ft}}^2$ .

Now, the big square mat has area ${\text{81 ft}}^2$ and is uses 4 identical rectangular mats and one small square mat. This gives that,

  $\begin{array}{c}4x+\frac{x}{2}=81\\ \frac{8x+x}{2}=81\\ 9x=2\left(81\right)\end{array}$

  $\begin{array}{c}9x=162\\ x=\frac{162}{9}\\ x=18\end{array}$

Conclusion:

Thus, the area of the rectangular mats is ${\text{18 ft}}^2$ .

b.

To determine

To find: the dimensions of the rectangular mat.

Answer to Problem 46E

Length = 6 feet

Width = 3 feet

Explanation of Solution

Given:

The area of the rectangular mat is ${\text{18 ft}}^2$ .

The length of rectangular mat is twice its width.

Let l be the length of the rectangular mat and w be the width of the rectangular mat.

Since, the length of rectangular mat is twice its width, it implies that:

  $l=2w$

Now, area of the rectangular mat is ${\text{18 ft}}^2$ .

  $\begin{array}{c}Area=length\times width\\ 18=l\times w\\ 18=2w\times w\end{array}$

  $\begin{array}{c}\frac{18}{2}=w^2\\ 9=w^2\\ 3=w\end{array}$

Then the length of rectangular mat is,

  $\begin{array}{c}l=2w\\ =2\times 3\\ =6\end{array}$

Conclusion:

The length of rectangular mat is 6 feet and its width is 3 feet.