Stuck on this problem

Question

A rancher has 100 yards of fence with which to enclose three sides of a rectangular field (the fourth side is a river and will not require fencing). Find the dimensions of the field with the largest possible area. (For the purpose of this problem, the width will be the smaller dimension (needing two sides); the length with be the longer dimension (needing one side).)

length =

width =

What is the largest area possible for this field?

area =

Answer 1

This is a classic optimization problem in Calculus. Define x as the length of sides perpendicular to river and y as the length of side parallel to river. Therefore 2x+y = 100 (yards) because that is all the fence.

Area is then A = x*y We want the area to be a function of a single variable. Here it is two variable so we look at the equation above and rewrite as y = 100-2x. Now sub in for y and you have a function for area based alone on x

A(x) = x*(100-2x) = 100x-2x^2

The max occurs where the derivative is zero. Notice the upside-down parabola.

A'(x) = 100 - 4x. If this is zero, then x = 25 and y therefore is 50. If you did not understand the derivative idea, then when you graphed the function, you would notice that it touches x axis at both 0 and 50. Because it is an upside-down parabola, it must be symmetrical so the middle is where x = 25.