A farmer has 200 feet of fencing to enclose a rectangular garden, using the fencing for all four sides. If A(w) = w(100-w) models the…

Expand the area model: $A(w) = w(100 - w) = 100w - w^2$. This is a downward-opening parabola, so the maximum occurs at the vertex: $w = -\frac{b}{2a} = -\frac{100}{2(-1)} = 50$ feet, choice B. A width of 50 means a length of $100 - 50 = 50$ as well, so the best rectangle is a square with area 2500 square feet. Choice A (100) would give zero area, since the length would be 0. Choice C (25) gives a smaller area, 1875. Choice D (200) is the total fencing and gives a negative length. The maximum comes from the vertex.