John wants to separate a rectangular part of his garden for his puppy. He has material for a 100-meter long fence and he plans to use one side of his house as a barrier. How should John select the sizes of his fence in order to gain the biggest territory?

We can denote the sides of the rectangle with a and b. As one side of the separated area is going to be the wall of the house, we do not have to use a fence there. Therefore, we will have two sides, one denoted by a and one denoted by b (as the other b is the wall). The territory of the separated area can be written as a function f(x)=ab (a multiplied by b). The circumference of the territory is a+a+b=100. Thus, b=100-2a. Now, if we plug this back into f(x)=ab we can express the territory as f(x)=a(100-2a)=100a-2a².Our task is to find the maximum value of function f(x)=100a-2a² which we can do by differentating it.The first derivative of f(x) is f'(x)=100-4a. Our first order condition to find the maximum of f(x) is f'(x) to be equal to 0. Setting 100-4a=0 we get that a=25.We can check whether it is local maximum or minimum point by differentiating the expression again for which we get f''(x)=-4. As the second derivative is a negative number, our second order condition is satisfied for a=25 being the maximum point.Plugging a=25 into the expression b=100-2a we get b=50.Thus, the sizes that give us the maximum territory is 25 meters and 50 meters respectively.The maximum territory is 25x50=1250.