Answer:
A. [tex]x=\frac{-7+\sqrt{193}}{4}[/tex]
Step-by-step explanation:
We have, the dimensions of the living room are 30 ft and 20 ft.
Thus, area of the living room = length × width = 30 × 20 = 600 ft².
Now, it is given that the cabinet takes 6% of the living room i.e. 6% of 600 ft² = 0.06 × 600 = 36 ft².
As, the triangle has dimensions (2x+3) ft and (3x+6) ft.
So, the area of the triangle = [tex]\frac{1}{2}\times base\times height[/tex]
i.e. Area of cabinet = [tex]\frac{1}{2}\times (2x+3)\times (3x+6)[/tex]
i.e. Area of cabinet = [tex]\frac{3}{2}\times (2x+3)\times (x+2)[/tex]
i.e. Area of cabinet = [tex]\frac{3}{2}\times (2x^2+7x+6)[/tex]
Since, the cabinet takes 6% of the living room, we have,
[tex]\frac{3}{2}\times (2x^2+7x+6)[/tex] = 36
i.e. [tex]2x^2+7x+6=36\times \frac{2}{3}[/tex]
i.e. [tex]2x^2+7x+6=24[/tex]
i.e. [tex]2x^2+7x-18=0[/tex]
Further, as the solution of a quadratic equation [tex]ax^{2}+bx+c=0[/tex] is given by [tex]x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}[/tex]
On comparing we have, a=2, b=7, c= -18.
Thus, [tex]x=\frac{-7\pm \sqrt{7^{2}-4\times 2\times (-18)}}{2\times 2}[/tex]
i.e. [tex]x=\frac{-7\pm \sqrt{49+144}}{4}[/tex]
i.e. [tex]x=\frac{-7\pm \sqrt{193}}{4}[/tex]
i.e. [tex]x=\frac{-7+\sqrt{193}}{4}[/tex] and [tex]x=\frac{-7-\sqrt{193}}{4}[/tex]
So, according to the options, we have,
A. [tex]x=\frac{-7+\sqrt{193}}{4}[/tex] is the correct value of x.
