



There
are lots of ways to ask volume problems.



The one
that seems to show up the most often is the box 





Example: 


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
= 4X^{3}  60X^{2} + 200X



Taking
the derivative: 


V'
= 12X^{2}  120X + 200



Solve for
V' = 0: 


0
= 12X^{2}  120X + 200



0 = 3X^{2}
 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
= AX^{2} + BX + C






You can
solve for X using the equation: 





So: 


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'
= 12X^{2}  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 
