© 2002 by Karl Hahn
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Step 1: Draw the diagram. It will help you visualize the problem. This one is pretty easy, so I won't show it here, but instead let you draw it. The rectangle should be divided into two smaller by a line segment that runs parallel to the highway. I am labeling the width, W, as the length of fence along the highway, and the length, L, as the dimension of the rectangle perpendicular to the highway. If you did the opposite, that's ok too.
Step 2: The area of the region within 25 meters of the highway is 25W. The area of the region more than 25 meters from the highway is (L - 25)W. So the total yield is
Y = 25Wb + 2(L - 25)Wb = (2L - 25)Wb
Step 3: The perimeter equation is:
600 = 2L + 2WI am going to solve for L in terms of W, but if you did it the other way (solving for W in terms of L), that works too.
L = 300 - W
Step 4: Substitute the expression for L into the yield equation.
Y = (2(300 - W) - 25)Wb = (-2W2 + 575W)b
Step 5: Find the derivative. We need the derivative of Y with respect to remaining independent variable, W. Remember that b is a constant.
dY= (-4W + 575)b dW
Step 6: Set the derivative to zero and solve for W.
0 = (-4W + 575)bThe b cancels, which is why you never needed to know what it is.
0 = -4W + 575 W = 143.75 meters
Step 7: Substitute the solved width back into the perimeter equation to get the length, L.
600 = 2W + 2L 600 = 287.5 + 2L 312.5 = 2L L = 156.25 meters