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

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,  $x = \frac{-7+ \sqrt{193} }{4}$:
$x= \frac{-b+ \sqrt{ b^{2} -4ac} }{2a} = \frac{-7+ \sqrt{ (7)^{2} -4*2*(-18)} }{2*2} = \frac{-7+ \sqrt{49+144} }{4} = \frac{-7+ \sqrt{193} }{4}$