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### Optimization

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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/
```
Associated Topics:
High School Calculus
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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