Find the area of a rectangle if its perimeter is 84 cm and the sides of the rectangle are directly proportional

Find the area of a rectangle if its perimeter is 84 cm and the sides of the rectangle are directly proportional to the numbers 3 and 4.

Let us denote the lengths of the sides of the rectangle through x and y.

According to the condition of the problem, the perimeter of this rectangle is 84 cm, therefore, we can write the following ratio:

2 * (x + y) = 84.

It is also known that the lengths of the sides of a given rectangle are directly proportional to the numbers 3 and 4, therefore, we can write the following ratio:

x / y = 3/4.

We solve the resulting system of equations.

Substituting into the first equation the value x = (3/4) * y from the second equation, we get:

2 * ((3/4) * y + y) = 84;

2 * (7/4) * y = 84;

(7/2) * y = 84;

y = 84 / (7/2);

y = 84 * (2/7);

y = 2 * 84/7;

y = 2 * 12;

y = 24 cm.

Knowing y, we find x:

x = (3/4) * y = (3/4) * 24 = 3 * 24/4 = 3 * 6 = 18 cm.

Answer: the lengths of the sides of this rectangle are 18 cm and 24 cm.