Homework #4
Instructions. This is due on October 30
at the beginning of class. It is worth 20 points. Late papers lose 3 points per
day or partial day late. You must work by yourself, but feel free to consult
with me. Explain what you did and why you did it. Your paper should start with
an introduction and finish with a conclusion. Start early, ask questions and do
well.
Introduction. This assignment guides you through a technique called the Lagrange multiplier method. It is used by engineers and economists as a general technique for finding optimal solutions to a variety of problems. In this problem, assume that a landowner’s property line runs along the path y = 6 - x. The landowner wants to run an irrigation ditch from a reservoir in the shape of the ellipse 4x2 + 9y2 = 36. The landowner wants to build the shortest ditch possible from the reservoir to the closest point on the property line.
Problems. Start by sketching a picture showing the property line and
reservoir and a ditch line that looks like the shortest ditch possible. Briefly
explain how you chose the closest point on the property line.
Start a new picture showing the property line and
reservoir. This time, sketch in a line that is tangent to the ellipse and
parallel to the property line. Argue that the ditch should start at the point
of tangency and run perpendicular to the two lines. How does this ditch compare
to the ditch you drew in the first step?
The next steps will produce the point on the ellipse
where the ditch starts. Use implicit differentiation to find the slope (in
terms of x and y) of a tangent to the ellipse. For this tangent
line to be parallel to the property line, it must have the same slope. So, set
the slope equal to -1. Solve this equation for x
and substitute this into the equation of the ellipse. Solve for y ; this
is the y-coordinate of the point we want. Find the corresponding x-coordinate.