Calculus Max Min Volume Problems
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Max Min Volume Problems


 There are lots of ways to ask volume problems.

 The one that seems to show up the most often is the box
 What is the maximum possible volume open top box that can be built
 from a flat sheet of stuff (maybe cardboard or steel) that is 10" by 20" ?


 The trick is to know that you build the box by cutting squares out of the corners 
 of the sheet of stuff, and folding the sides up.
 For this to work, ALL of the cuts need to be the same size.


 The equation for the volume of a box is:


 Volume = Length x Width x Height

 So calling the length of all of our cuts X, we can write this as:

 Volume = (20 - 2X)(10 - 2X)X

 Multiplying this out, we get:

 V = 4X3 - 60X2 + 200X

 Taking the derivative:

 V' = 12X2 - 120X + 200

 Solve for V' = 0:

 0 = 12X2 - 120X + 200

0 = 3X2 - 30X + 50

 This puppy isn't going to factor very easily.
 We will need to use the Quadratic Formula.
 Do you remember it?
 It goes like this:
 When you have an equation of the form:

 0 = AX2 + BX + C

 You can solve for X using the equation:


 A = 3, B = -30, and C = 50


 Now that we have answers,
 we have to check to see if they make any sense in our problem.
 The short side of the piece of stuff is 10" long.
 That means we can't cut two 7.9" sections out of it.
 So that answer won't work in our problem.
 But the 2.1 answer will.
 We also do have endpoints.
 We could cut nothing from the corners, so X would equal zero.
 Or we could cut half of the short side from each of the corners,
 so X would equal 5.
 You can probably see that if we do either one of these endpoint plans,
 we won't get much of a box.
 But if the problem was more complex, we might not be able to see that right off.
 In any case, we will check all three suspect points.
 If H = 0, 
 then L = 20 - 2x0 = 20
 and W = 10 - 2x0 = 10
 That makes the volume:

 Volume = 20 x 10 x 0

 Volume = 0

 If H = 5, 
 then L = 20 - 2x5 = 10
 and W = 10 - 2x5 = 0
 That makes the volume:

 Volume = 10 x 0 x 5

 Volume = 0

 If H = 2.1, 
 then L = 20 - 2x2.1 = 15.8
 and W = 10 - 2x2.1 = 5.8
 That makes the volume:

 Volume = 15.8 x 5.8 x 5

 Volume = 192.4

 Looks like a winner!
 BUT ...
 Let's check the second derivative to make sure it's a maximum.
 Yes, I know it HAS to be, but humor me.
 Starting with the equation for the first derivative:

 V' = 12X2 - 120X + 200

 V'' = 24X - 120

 Solve for X = 2.1 ...

 V'' = 24X - 120

 V'' = 24(2.1) - 120

 V'' = 50.4 - 120

 V'' = -69.6

 V'' is negative so the graph is concave down.
 That means that X = 2.1 is a relative maximum point.
 Since we have checked the endpoints and they give you smaller areas,
 we know that X = 2.1 is the absolute maximum.
 And the height value of our answer.

   copyright 2005 Bruce Kirkpatrick

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