# Math Help - Volume and Surface Area

1. ## Volume and Surface Area

I have this assignment and I find it really hard. I am a Filipino and just moved her in Australia thats why i find it really hard understanding this questions. Please help me. This is the question. "Your aim in this question is to design a cylindrical coffee cup so that the dimensions give the best surface area to volume ratio. for your coffee to stay hot the coffee cup should hav as small a surface area to volume ratio as possible. For this question assume a particular volume (between 150mL and 300mL. Your answer should be based on results shown in spreadsheet tables and graphs." I dont get this at all (. Please help.

2. You've got a cylinder. Let's call the radius r, and the height h. Can you write a formula for the volume? How about for the surface area?

3. volume for cylinder is Pi*r^2*h .. and surface area is 2*Pi*r*h + 2*Pi*r^2
is that right? sorry dont really know how to write formulas in here

4. Correct. So your surface area to volume ratio is given by

$\frac{A}{V}=\frac{2\pi rh+2\pi r^{2}}{\pi r^{2}h}=2\,\frac{rh+r^{2}}{r^{2}h}.$

This is what you want to minimize, subject to holding the volume fixed. The problem is that you're allowed to vary both the radius and the height. You'd like to reduce the problem down to minimizing this ratio by only changing one parameter. Well, you can do that. You can eliminate either r or h. Off-hand, I would say that you should use the volume equation to eliminate h, and write this ratio in terms of r and V only (remember that V is constant!). What does that give you?