The area of the rectangle is 30 cm2 and the perimeter is 26 cm, what is the length of its sides?
Let’s denote the sides of the rectangle by x cm and y cm.
Let’s write down what the perimeter of this rectangle is.
2x + 2y = 26.
Let’s write down what is the area of the rectangle.
x * y = 30.
Let’s compose a system of equations and solve it.
2x + 2y = 26,
x * y = 30.
Divide the first equation by 2.
x + y = 13,
x * y = 30.
Find x from the first equation.
x = 13 – y,
x * y = 30.
Substitute the first equation into the second.
(13 – y) * y = 30.
13y – y ^ 2 = 30.
13y – y ^ 2 – 30 = 0.
-y ^ 2 + 13y – 30 = 0.
We got a quadratic equation, multiply both sides by (- 1) and solve it.
y ^ 2 – 13y + 30 = 0.
D = b ^ 2 – 4ac = 169 – 4 * 1 * 30 = 49> 0.
y1 = (13 – 7): 2 * 1 = 3.
y2 = (13 + 7): 2 * 1 = 10.
x1 = 13 – y1 = 13 – 3.
x1 = 10
x2 = 13 – y2 = 13 – 10.
x2 = 3.
Hence, the sides of the rectangle are 3 cm and 10 cm, or 10 cm and 3 cm.
Answer: The sides of the rectangle are 3 cm and 10 cm, or 10 cm and 3 cm.