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:
[tex]a + 2b = 25 \implies a = 25-2b[/tex]
And we want to maximize a x b.
Filling in the expression for a in the area, we actually want to maximize:
[tex](25 - 2b) \cdot b = -2 b^{2} + 25b[/tex]
This is a mountain parabola. To find its maximum, we equal the derivative to 0:
[tex]25 - 4b = 0 \implies b=6.25[/tex]
From this follows a = 12.5, but now we have proven that C is really the optimum!
