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/
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