A 100m long fence is required to enclose a rectangular area of the largest area. Find the dimensions of this site, area and perimeter.

Divide the length of the fence by 2, this will be the sum of the length of the rectangular area and its width.
p = 100 m ÷ 2 = 25 m.
Let’s designate the length as x, then the width will be 50 m – x.
Let’s compose and solve the equation.
S = x * (50 m – x) = 50 m * x – x².
Consider the function S (x) = 50 * x – x².
Take the derivative of the function.
S ‘(x) = 50 – 2 * x.
Let us equate the derivative of the function to 0.
50 – 2 * x = 0.
x = 25.
Take the values of the function at the edges of the segment [0; 50]
S (50) = 50 * 50 – 50² = 0.
S (0) = 50 * 0 – 0² = 0.
S (25) = 50 * 25 – 25² = 625 – max.
Let’s find the width.
50 m – 25 m = 25 m.
Answer: the size of the site is 25 m by 25 m.