# A 4m by 6m rug covers half of the floor area of a room, and leaves a uniform strip of bare floor around the edges. What are the dimensions of the rooms?

Oct 6, 2015

The room is 6 m by 8 m

#### Explanation:

Room has an area of $2 \times \left(4 \times 6\right) = 48 {m}^{2}$
(since the rug covers half the area and the rug has an area of $4 \times 6 {m}^{2}$)

Let
$\textcolor{w h i t e}{\text{XXX}} L =$ length of the room
$\textcolor{w h i t e}{\text{XXX}} W =$ width of the room
$\textcolor{w h i t e}{\text{XXX}} s =$width of the uniform strip around the outside of the rug

$L = 6 + 2 s$
$W = 4 + 2 s$

Area of room
$L \times W = \left(6 + 2 s\right) \left(4 + 2 s\right) = 48$

$\rightarrow 24 + 20 s + 4 {s}^{2} = 48$

$\rightarrow 4 {s}^{2} + 20 s - 24 = 0$

$\rightarrow {s}^{2} + 5 s - 6 = 0$

$\rightarrow \left(s + 6\right) \left(s - 1\right) = 0$

$\rightarrow s = - 6 \text{ or } s = 1$

Since a strip width of $\left(- 6\right)$ is impossible
the strip has a width of $1$ and
$L = 6 + 2 s = 8$
and
$W = 4 + 2 s = 6$