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

A.
`x=(-7+√(193))/(4)`

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 =
`(1)/(2)* base* height`

i.e. Area of cabinet =
`(1)/(2)* (2x+3)* (3x+6)`

i.e. Area of cabinet =
`(3)/(2)* (2x+3)* (x+2)`

i.e. Area of cabinet =
`(3)/(2)* (2x^2+7x+6)`

Since, the cabinet takes 6% of the living room, we have,

`(3)/(2)* (2x^2+7x+6)` = 36

i.e.
`2x^2+7x+6=36* (2)/(3)`

i.e.
`2x^2+7x+6=24`

i.e.
`2x^2+7x-18=0`

Further, as the solution of a quadratic equation
`ax^(2)+bx+c=0` is given by
`x=\frac{-b\pm \sqrt{b^(2)-4ac}}{2a}`

On comparing we have, a=2, b=7, c= -18.

Thus,
`x=\frac{-7\pm \sqrt{7^(2)-4* 2* (-18)}}{2* 2}`

i.e.
`x=(-7\pm √(49+144))/(4)`

i.e.
`x=(-7\pm √(193))/(4)`

i.e.
`x=(-7+√(193))/(4)` and
`x=(-7-√(193))/(4)`

So, according to the options, we have,

A.
`x=(-7+√(193))/(4)` is the correct value of x.