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=(-7+√(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=(1)/(2)* b* h=(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* 20=600~\textup{sq. feet.}`

According to the question,

`a=6\%* A\\\\\Rightarrow (1)/(2)(2x+3)(3x+6)=(6)/(100)* 600\\\\\Rightarrow (2x+3)(3x+6)=24* 3\\\\\Rightarrow (2x+3)(x+2)=24\\\\\Rightarrow 2x^2+7x-18=0\\\\\Rightarrow x=(-7\pm√(7^2-4* 2*(-18)))/(2* 2)\\\\\\\Rightarrow x=(-7\pm√(193))/(4).`

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

`x=(-7+√(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.