The perimeter of the rectangle is 56cm. What are the sides if this rectangle has the largest area?

To solve the problem, we will use the formulas for the perimeter and area of a rectangle:
P = 2 * (a + b).
S = a * b.
By the condition of the problem, the perimeter is 56 cm.
2 * (a + b) = 56
a + b = 28.
Let a = x, then b = 28 – x.
Consider the area of the rectangle as a function of x.
S (x) = x * (28 – x) = 28x – x².
Find the derivative S ‘(x).
S ‘(x) = 28 – 2x = 2 * (14 – x);
S ‘(x) = 0;
2 * (14 – x) = 0
14 – x = 0
x = 14 (cm)
Side a was taken as x, side b = 28 – x = 28 – 14 = 14 (cm).
We got that the rectangle given by the condition is a square with a side of 14 cm.
Answer: the largest area will be if the sides of the rectangle are 14 cm and 14 cm.