Optimization

Date: 11/26/96 at 21:51:37
From: Sanec
Subject: Optimization word problem

Dr. Math:
I am a senior in an AP calculus class.  We just finished max/min 
problems, first and second derivative tests, and implicit 
differentiation.  Now we are on optimization problems.  Would you 
please explain how I would go about solving the following problem?:

To make a funnel, we take a circular piece of metal, cut out a sector, 
and connect the two radial edges together to make an open cone.  What 
should the angle of the sector be to maximize the volume of the cone?

Thank you for your time.

Date: 11/29/96 at 13:22:12
From: Doctor Keith
Subject: Re: Optimization word problem

Hi,

Great problem.  I am taking a graduate course on this so I know how 
you are feeling.  Optimization problems can get very tricky, but they 
are very useful to know how to do and can even be fun!  I will be 
following this pattern that is a good way to work with any 
optimization problem: 

1) Clarify problem
2) Find what you need to know in order to do optimization
3) Use other areas of math (trig, algebra, etc) to get the problem
   into a form that will allow direct application of the standard
   derivative conditions
4) Perform differentiation and apply rules to get optimum value (for
   you to do!)

-----------------
1) Let's review what we have and what we want:

-> a circular piece of metal, say of radius R 
-> the ability to cut out a sector of the circle, say of angle A

-> we want a funnel (cone) with the largest possible volume

2) A good place to start is the equation for the volume of a cone, 
which is:

       vol=(1/3)pi*r*r*h
       where  r is the radius of the base (not R)
              h is the height of the cone

So we need to express the volume as a function of A so you can apply 
those first and second derivative tests to find the maximum volume 
(i.e., you will have optimized it).  To do this we need to get
r and h as functions of A.

3) First let's try to get r.  I suggest you draw a picture of what we
have, both the metal sheet and the cone, since it will make things 
easier.  If you have problems, write back and I will draw something up 
on my computer.  

Okay, if we look at the base of the cone, we know that the 
circumference is 2pi*r.  But the circumference of the base is the 
circumference of the circular sheet of metal - the length of the edge 
of the sector we removed. Since we are measuring angles in radians, 
the second calculation is just (2pi-A)R. Note that  2pi*R is the 
original circumference and A*R is the length we got by removing a 
sector of angle A (in radians). Thus we can equate and get:

      2pi*r=(2pi-A)R
or
          r=(1-A/2pi)R

and we have our formula for r.

Now we need h in terms of R and A.  Note that R,h, and r form a right 
triangle with R as the hypotenuse, so we can use the Pythagorean 
theorem to get (note ^ is used for powers):

         h^2 + r^2 = R^2
         h^2 = R^2 - r^2
             = R^2 - ((1-A/2pi)^2)*R^2
             = (1 - (1 - A/pi - (A^2)/4(pi^2)))R^2
             = ((4pi*A-(A^2))/(4(pi^2)))R^2

Thus
         h   = (sqrt(4pi*A-(A^2))/2pi)R

So now you can substitute in for r and h and get an equation for the
volume of the funnel in terms of R (a constant) and A (the only 
variable).  At this point you can use your derivative conditions to 
find the solution.

Like most optimization problems, this problem shows that a lot of the 
work is spent getting the problem into form.  Applying the procedure 
to determine the optimum (in this case the largest) is usually the 
easiest part of the problem.  If you have more problems like this or 
you would like further clarification or a picture, just write back and 
I would be happy to help.  Hope you find this useful.

Good luck.

-Doctor Keith,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/