Question

Leonhard wants to place a triangular-based cabinet in the corner of his rectangle-shaped living room. The triangular base has a length of 2x + 3 feet and a height of 3x + 6 feet. What value of x causes the cabinet to take up 6% of the living room floor?

Answer 1

Answer: The answer is $(A)~~x=\dfrac{-7+\sqrt{193}}{4}.$

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

$b=(2x+3)~\textup{feet},\\\\h=(3x+6)~\textup{feet}.$

Therefore, area of the triangular cabinet is given by

$a=\dfrac{1}{2}\times b\times h=\dfrac{1}{2}(2x+3)(3x+6)~\textup{sq. feet.}$

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

$A=30\times 20=600~\textup{sq. feet.}$

According to the question,

$a=6\%\times A\\\\\Rightarrow \dfrac{1}{2}(2x+3)(3x+6)=\dfrac{6}{100}\times 600\\\\\Rightarrow (2x+3)(3x+6)=24\times 3\\\\\Rightarrow (2x+3)(x+2)=24\\\\\Rightarrow 2x^2+7x-18=0\\\\\Rightarrow x=\dfrac{-7\pm\sqrt{7^2-4\times 2\times(-18)}}{2\times 2}\\\\\\\Rightarrow x=\dfrac{-7\pm\sqrt{193}}{4}.$

Since the length of something cannot be negative, so the value of 'x' is

$x=\dfrac{-7+\sqrt{193}}{4},$

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.