July 18, 2013†††††††††††††††††††††††††††††††††††††† Name:† ††††††††††††††††††††††† Key†††††††††††††††††††††††††††††††††††††††††††††††††††††
This assignment will count as a quiz.† It is due at the beginning of class on Tuesday, July 24.
You may use your book and your notes, and may even receive help from others, but all work you turn in must be your own.†
Answers must be fully justified to earn credit!
For each of the following problems,†
a.† Find the objective function
b.† Find the constraint equation(s), and then
(In addition to the steps above, you will likely need to perform other steps, such as drawing a picture, introducing variables, etc.)
1.† A farmer has 2400 feet of fence.† He needs to make 2 rectangular pens that share a border (picture a rectangle with a divider down the middle).†† Help the farmer determine what dimensions will produce the maximum possible total area.
2.† The farmer owns a very long field - it is 1 km wide and 2 km long.† He has roads on each side of the field, but no roads which cut through it.† He needs to walk from one corner of the field to the opposite corner.† His speed is 5 km/hr when walking along the road, but only 3km/hr when walking through the muddy field.† What is the quickest route?† How long will it take him?† (note:† you may safely assume that the optimal route consists of at most 2 legs: one along the road and one through the field.)
††††††††††††††††††††††† 2-x†††††††††††††††††† †††††† x
††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ††††† 1
If T = the time required, then
Setting Tí(x) = 0, we get
So the only critical point of this function on the domain [0,2] is when x=3/4, at which point the walk requires 2/3 of an hour, or 40 minutes. Comparing this to the endpoints: when x=2, the farmer walks diagonally through the mud, for a time of
According to the formula, if T=0 the time required would be 2/5† +1/3 hour, or 44 minutes, but given the road available on the far side, this would really require 3 km of walking along roads, for a total time of 3/5 hours, or 36 minutes.