Wednesday, September 2, 2009

When Does This Become Useful?!

Math Problem...
Surface Area of a cylinder = 2*pi*radius^2+2*pi*radius*height
Volume of the cylinder = 300cm^3 or pi*radius^2*height
How do i find the least possible values for the radius and height of the cylinder?

What am I trying to minimize, the surface area?
The radius and height are inversely related (by your constraint: height = volume / (pi * radius^2)) so I can't minimize them both. In fact, I could let the radius go to zero and the height go to infinity and still have a finite volume (in the limit). The same applies for zero height and infinite radius in the limit. Assume that I'm trying to minimize the surface area.

The trick is to express what I'm trying to minimize in terms of a single variable.
Let S = surface area, V = volume, r = radius, h = height. So:

S = 2 * pi * r^2 + 2 * pi * r * h
V = pi * r^2 * h = 300

Let's put S in terms of r by eliminating h:

h = 300 / (pi * r^2)

  • S = 2 * pi * r^2 + 2 * pi * r * 300 / (pi * r^2) = 2 * pi * r^2 + 600 / r
  • Now S is just a function of r which you can minimize in the usual way:
    dS/dr = 0, solve for r. With r you can then find h and S.
Is that right?! Anyone? Anyone?

Advanced Biology...
How does a water strider walk on water?
How does a bat navigate through a cave or stay warm at night?
How does a bumble bee fly? How does a gecko walk on the ceiling?
How does a tree draw water up to its highest leaves?
How do nutrients pass through the walls of a cell?
What determines the form of a given species?
The fact that all biology problems pertain to life is secondary to the application of physics for the purpose of solving them. Understanding physics properties, and in particular understanding how size determines which physics properties are most relevant, is the key to answering the majority of biology questions.

I think somewhere in the "REAL" world I will need this stuff. Or at least that's what I'm being brainwashed to believe. Between school, homework, jobs, church commitments, working out, coaching, chores, social commitments, blah blah blah blah blah ...supposedly LIFE is happening.

Oh right, right. lol.