Afarmer has 25 yards of fencing to make a pig pen. he is going to use the side of the barn as one of the sides of the fence, so he only needs to fence 3 sides. what should be the dimensions of the fence in order to maximize the area?
4 yards by 17 yards
b) 5 yards by 15 yards
c) 6.25 yards by 12.5 yards
d) 13.25 yards by 13.5 yards

By simply calculating the areas you can easily find that C is the largest (78.125). But let's approach this one mathematically:

If the side of the fence opposite of the barn has length a, and the sides perpendicular to the barn has length b, we know that:



And we want to maximize a x b.

Filling in the expression for a in the area, we actually want to maximize:



This is a mountain parabola. To find its maximum, we equal the derivative to 0:



From this follows a = 12.5, but now we have proven that C is really the optimum!