Among the rectangles, the sum of the lengths of three sides of which is 20 cm

Among the rectangles, the sum of the lengths of three sides of which is 20 cm, find the perimeter of the straight line with the largest area.

Let’s denote the length and width of the rectangle as x and y.

y + 2x = 20;

y = 20 – 2x;

The area of a rectangle is equal to the product of its length and width:

s = x * (20 – 2x) = -2x ^ 2 + 20x;

Consider the area of a rectangle s as a function of its length x. The function will have a maximum at the point where its derivative is zero:

s’ = -4x + 20;

-4x + 20 = 0;

x = 5;

Let’s find the width of the rectangle and its perimeter:

y = 20 – 2 * 5 = 10 cm;

P = (5 + 10) * 2 = 30 cm.

Answer: P = 30 cm.