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
[tex]= (27-2x)(19-2x)[/tex]
[tex]= 513-54x-38x+4x^2[/tex]
[tex]= 513-92x+4x^2[/tex]
According to the question,
[tex]513-92x+4x^2=153[/tex]
[tex]513-92x+4x^2-153=0[/tex] ( subtract 153 both sides)
[tex]4x^2-92x+360=0[/tex]
[tex]4(x^2-23x+90)=0[/tex]
[tex]x^2-23x+90=0[/tex]
[tex]x^2-(18+5)x+90=0[/tex] ( Middle term splitting)
[tex]x^2-18x-5x+90=0[/tex]
[tex]x(x-18)-5(x-18)=0[/tex]
[tex](x-5)(x-18)=0[/tex]
[tex]x-5=0[/tex] or [tex]x-18=0[/tex] ( zero product property)
[tex]x=5[/tex] or [tex]x=18[/tex]
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.
