The perimeter of the rectangle is 30 cm and its area is 54 cm squared. What are the length and width of the rectangle?

1. We take the length of the rectangle for x, for its width.
2. Let’s compose two equations:
2x + 2y = 30;
xy = 54; x = 54 / y.
3. Substitute x = 54 / y in the first equation:
108 / y + 2y = 30;
(108 + 2y ^ 2) / y = 30;
2y ^ 2 – 30y + 108 = 0;
y ^ 2 – 15y + 54 = 0;
The equation has 2 solutions.
The first value y = (15 + √15 ^ 2 – 4 x 54) / 2 = (15 + 3) / 2 = 9 cm; x = 54: 9 = 6 cm.
The second value is y = (15 – 3) / 2 = 6 cm; x = 54: 6 = 9 cm.
Answer: the length must be greater than the width, so the length of the rectangle is 9 cm, width is 6 cm.