A rectangular dog pen is constructed using a barn wall as one side and 50 meters of fencing for the other three sides. What is the maximum area of the dog pen?

Answer 1

length of side parallel to barn === y
length of each of other two sides === x
so we have 2x + y = 50
y = 50-2x

area = xy = x(50-2x)
= -2x^2 + 50x

We need the vertex of this parabola
the x of the vertex = -50/-4 = 12.5
then y = 50-2(12.5) = 25

so max area = xy = 12.5(25) m^2 or 312.5 m^2

You could also find x by completing the square,
or by Calculus setting the derivative of -2x^2 + 50x equal to zero