Math 201: Applied Optimization (4.6) Day 44
To optimize quantities in real world situations we can follow the steps below.
i) Determine what quantity you want to optimize and give it a name (call it yfor now).
ii) Write yas a function of other quantities in the problem.
iii) Use other information in the problem to re-write yas a function of only one variable.
iv) Find all local extrema of y.
v) Find the behavior of ynear the edges of the domain.
On a closed interval [a,b] this means finding the value of yat aand b.
On an open interval (a, b) this means finding the limit of yas it approaches aand b.
(In the open interval aand bcould be ±∞.)
vi) Compare the yvalues from the previous two parts to determine absolute extrema, if they exist.
vii) Interpret your answer and state a solution to the problem given
Use these steps to solve the following problems.
(1) The legs of a right triangle (not including the hypotenuse) have a total length of 12. What
is the maximum area that this triangle could have?
(2) The United States Postal Service considers a rectangular package “oversized” if the sum
of the length, width, and height is more than 108 inches. If the length is twice the width,
what is the largest volume that a package can be without being “oversized?” What are the
dimensions of the box?
(3) Speedy the mouse can run 8mph and swim 2mph. Speedy wants to cross a river that is
1/10 of a mile wide and end up 1/5 mile downstream. If Speedy swims in a diagonal line
across the river and then runs down the river bank, what is the minimum amount of time
that it could take Speedy to get to the destination? How far will Speedy swim? How far
will Speedy run? What is the total distance Speedy will travel?
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