If If we are limited to 950 ft of fencing, what dimensions of fence should be used to maximize the area?

Question

A fence is to be constructed in a rectangular shape partitioned into three equal parts. If we are limited to 950 ft of fencing, what dimensions of fence should be used to maximize the area?

Answer 1

A = -2W² + 475W

From Algebra maximum occurs at -475 / (2)(-2)

Answer 2

If I understand the question correctly, the formula for the amount of fencing isn't just the perimeter because you have it sectioned off. So it would be: 4W + 2L = 950 (where W is width and L is length, you can use x and y or whatever you want). By solving that equation for L, we get: L = 475 - 2W. Then you have the basic area formula: A = LW. By substituting the length equation into the area equation, you get: A = (475 - 2W)(W), or A = 475W - 2W². If you remember back to algebra, that's just an upside down parabola if you were to graph it. And that's the thing you want to maximize, so in other words, find the vertex of the parabola. The derivative of a function is the slope of the tangent line, which is equal to zero at local extrema (maxima and minima), so if you find the derivative of the area equation and set it equal to zero, you should be able to solve for W. I hope this helps, let me know if you have any questions.