The shorter side of a rectangle is 18 cm less than the longer side. The numerical value of its area is equal to 6 times the numerical value of its perimeter. What is the length (in cm) of its longer side?​

Correct option is D

Given:

Let the length of the longer side of the rectangle be L cm.

The shorter side is 18 cm less than the longer side, so the shorter side is (L - 18) cm.

The area of the rectangle is equal to 6 times the perimeter of the rectangle.

Formula Used:

The area of the rectangle is given by: A = L × (L - 18)

The perimeter of the rectangle is given by: P = 2(L + (L - 18)) = 4L - 36

The area is equal to 6 times the perimeter: A = 6 × P

Substituting the formulas for area and perimeter: L × (L - 18) = 6 × (4L - 36)

Solution:

First, expand both sides of the equation:

L × (L - 18) = 6 × (4L - 36)

L² - 18L = 24L - 216

Bring all terms to one side of the equation:

L² - 42L + 216 = 0

L² - (36+6)L + 216 = 0 

L² - 36L -6L + 216 = 0  

L( L - 36) - 6( L - 36) = 0 

(L-36)(L-6) = 0 

L = 36 , L = 6 

Since the shorter side would be negative if L = 6, the valid solution is L = 36.