Answer: The answer is [tex](A)~~x=\dfrac{-7+\sqrt{193}}{4}.[/tex]
Step-by-step explanation: Given that Leonhard wants to place a triangular based cabinet in the corner of his room which is rectangular in shape.
The base and height of the triangular cabinet are
[tex]b=(2x+3)~\textup{feet},\\\\h=(3x+6)~\textup{feet}.[/tex]
Therefore, area of the triangular cabinet is given by
[tex]a=\dfrac{1}{2}\times b\times h=\dfrac{1}{2}(2x+3)(3x+6)~\textup{sq. feet.}[/tex]
Now, the length and breadth of the rectangular room are 30 feet and 20 feet respectively. Therefore, the area of the room is given by
[tex]A=30\times 20=600~\textup{sq. feet.}[/tex]
According to the question,
[tex]a=6\%\times A\\\\\Rightarrow \dfrac{1}{2}(2x+3)(3x+6)=\dfrac{6}{100}\times 600\\\\\Rightarrow (2x+3)(3x+6)=24\times 3\\\\\Rightarrow (2x+3)(x+2)=24\\\\\Rightarrow 2x^2+7x-18=0\\\\\Rightarrow x=\dfrac{-7\pm\sqrt{7^2-4\times 2\times(-18)}}{2\times 2}\\\\\\\Rightarrow x=\dfrac{-7\pm\sqrt{193}}{4}.[/tex]
Since the length of something cannot be negative, so the value of 'x' is
[tex]x=\dfrac{-7+\sqrt{193}}{4},[/tex]
because if we take the other value of 'x', the values of (2x+3) and (3x+6) will become negative which is not possible.
Thus, (A) is the correct option.
