Answer:
The dimension of the rug would be 17 ft × 9 ft.
Step-by-step explanation:
Given
length of the room = 27 ft.
width of the room = 19 ft.
suppose, she leaves a uniform strip of x ft. around the rug.
So,
The length of rug = (27-2x)ft.
Width of rug= (19-2x)ft.
∴ Area of the rug= length×width
![= (27-2x)(19-2x)](https://tex.z-dn.net/?f=%3D%20%20%2827-2x%29%2819-2x%29)
![= 513-54x-38x+4x^2](https://tex.z-dn.net/?f=%3D%20%20513-54x-38x%2B4x%5E2)
![= 513-92x+4x^2](https://tex.z-dn.net/?f=%3D%20%20%20513-92x%2B4x%5E2)
According to the question,
![513-92x+4x^2=153](https://tex.z-dn.net/?f=513-92x%2B4x%5E2%3D153)
( subtract 153 both sides)
![4x^2-92x+360=0](https://tex.z-dn.net/?f=4x%5E2-92x%2B360%3D0)
![4(x^2-23x+90)=0](https://tex.z-dn.net/?f=4%28x%5E2-23x%2B90%29%3D0)
![x^2-23x+90=0](https://tex.z-dn.net/?f=x%5E2-23x%2B90%3D0)
( Middle term splitting)
![x^2-18x-5x+90=0](https://tex.z-dn.net/?f=x%5E2-18x-5x%2B90%3D0)
![x(x-18)-5(x-18)=0](https://tex.z-dn.net/?f=x%28x-18%29-5%28x-18%29%3D0)
![(x-5)(x-18)=0](https://tex.z-dn.net/?f=%28x-5%29%28x-18%29%3D0)
or
( zero product property)
or ![x=18](https://tex.z-dn.net/?f=x%3D18)
if x=18, dimension would be negative ( Not possible)
Thus, x= 5
Hence,
length of rug= 27-10=17 ft.
width of rug= 19-10=9 ft.