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 2% of the living room floor?

The dimension of the living room floor is not stated, but I found a similar question where the dimension of the living room floor is given as 45 feet x 60 feet.

Area of the living room floor = 45 * 60 square feet

= 2700 square feet

2% of the living room floor = 2/100 * 2700

= 54 square inches

Area of triangle = 1/2 (base * height)

Area of the triangular-based cabinet

Base = 2x + 3

height = 3x + 6

Area = 1/2 (2x + 3) (3x + 6)

= 1/2 (6x^2 + 21x + 18)

If the triangular-based cabinet will take up 2% of the living room floor

Then, area of triangular-based cabinet = 2% of the living room floor

⇒ 1/2 (6x^2 + 21x + 18) = 54

⇒ 6x^2 + 21x + 18 = 108

⇒ 6x^2 + 21x + 18 - 108 = 0

⇒ 6x^2 + 21x - 90 = 0

We have a quadratic equation, 6x^2 + 21x - 90 = 0

Solve for x

6x^2 + 21x - 90 = 0

Divide through by 3

(6x^2) / 3 + (21x/3) - (90/3) = 0

⇒ 2x^2 + 7x - 30 = 0

⇒ 2x^2 + 7x - 30 = 0

⇒ 2x^2 + 12x - 5x - 30 = 0

⇒ 2x (x + 6) - 5 (x + 6) = 0

Factorize

(2x - 5) (x + 6) = 0

∴ 2x - 5 = 0, OR x + 6 = 0

⇒ 2x = 5 OR x = - 6

⇒ x = 2.5 OR x = - 6

Therefore, the value of x which will cause the cabinet to take up 2% of the living room floor is 2.5 or - 6