# A rug is to fit in a room so that a border of consistent width is left on all four sides. If the room is 9 feet by 25 feet and the area of the rug is 57 square feet, how wide will the border be?

Aug 18, 2016

The width of the border is $3 '$.

#### Explanation:

Let the width of the border be $w$ feet.

Since the room is $25 '$ long, the length of the rug must be

25-w (i.e, width of border on one end)-w (i.e., width of border on the other end)$= \left(25 - 2 w\right)$

On the same lines, the breadth of the rug must be $\left(9 - 2 w\right)$.

Hence, the area of the rug$= \left(25 - 2 w\right) \left(9 - 2 w\right)$, given to be $57$.

$\therefore \left(25 - 2 w\right) \left(9 - 2 w\right) = 57$

$\therefore 225 - 68 w + 4 {w}^{2} - 57 = 0$

$\therefore 4 {w}^{2} - 68 w = 57 - 225 = - 168$. Completing the square on
the L.H.S., we have,

$\therefore 4 {w}^{2} - 68 w + {17}^{2} = 289 - 168 = 121$

$\therefore {\left(2 w - 17\right)}^{2} = {11}^{2}$

$\therefore 2 w - 17 = \pm 11$

$\therefore 2 w = 17 \pm 11 = 28 , \mathmr{and} , 6$

$\therefore w = 14 , \mathmr{and} , 3$

But, $w = 14$ will make the breadth of the rug$= - 19$, which is not possible.

Hence, the width of the border is $3 '$.