The perimeter of the rectangle is 24 cm, and its area is 32 cm2. Determine what is the length and width of the rectangle?

We know the perimeter and area of ​​the rectangle. In order to find the length and width of this rectangle, we will compose and solve a system of equations.

S = a * b;

P = 2 (a + b);

System of equations:

a * b = 32;

2 (a + b) = 24.

System of equations:

ab = 32;

a + b = 12;

Let us express the variable b from the second equation in terms of a.

System of equations:

ab = 32;

b = 12 – a.

We substitute the expression from the second instead of b in the first equation.

a (12 – a) = 32;

b = 12 – a.

12a – a ^ 2 – 32 = 0;

a ^ 2 – 12a + 32 = 0;

D = b ^ 2 – 4ac = 144 – 128 = 16;

a1 = (12 + 4) / 2 = 16/2 = 8;

a2 = (12 – 4) / 2 = 8/2 = 4.

A collection of systems.

System 1:

a = 8;

b = 12 – 8 = 4.

System 2:

a = 4;

b = 12 – 4 = 8.

Answer: 4 cm, 8 cm sides of the rectangle.