The answer is \(x = \frac{-7+ \sqrt{193} }{4}\).
There is very similar question on Brainly, in which dimensions of the living room are given (20 feet, 30 feet).
So, knowing this, we first need to calculate how much 6% of the living room is.
The area of the living room is 600 square feet, since it is rectangle-shaped:
P = a b = 200 feet 30 feet = 600 square feet.
6% of 600 square feet is 36 square feet:
600 square feet : 100% = A : 6%
A = 600 * 6 / 100 = 36 square feet
The area of the triangular-based cabinet (A) is \(A = \frac{a*h}{2}\), where a is the base, and h is the height of the triangle.
It is given:
A = 36
a = 2x + 3
h = 3x + 6
Now, let's implement this to the formula \(A = \frac{a*h}{2}\):
\(36 = \frac{(2x+3)(3x+6)}{2}\)
⇒ \(36*2=6 x^{2} +12x+9x+18\)
\(72=6 x^{2} +21x+18\)
Let's both sides divide by 3:
\(24=2 x^{2} +7x+6\)
⇒ \(2 x^{2} +7x-18=0\)
This is the quadratic formula (\(a x^{2} +bx+c=0\)) for which
\(x = \frac{-b+\sqrt{b^{2} -4ac} }{2a}\) or
\(x = \frac{-b-\sqrt{b^{2} -4ac} }{2a}\).
We will use only \(x = \frac{-b+\sqrt{b^{2} -4ac} }{2a}\), because the length cannot have negative value.
From the equation \(2 x^{2} +7x-18=0\), we know that
a = 2
b = 7
c = -18
Thus, \(x = \frac{-7+ \sqrt{193} }{4}\):
\(x= \frac{-b+ \sqrt{ b^{2} -4ac} }{2a} = \frac{-7+ \sqrt{ (7)^{2} -4*2*(-18)} }{2*2} = \frac{-7+ \sqrt{49+144} }{4} = \frac{-7+ \sqrt{193} }{4}\)
