A Sphere's Surface Area and VolumeDate: 12/17/98 at 22:21:53 From: Matt Subject: Relation between a sphere's surface area and its volume What is the relation between a sphere's surface area and its volume? I plotted a graph of (4 (pi)r^2)/(4/3 (pi)r^3), which is surface area/ volume. I need to know how this helps, if at all. Date: 12/18/98 at 08:17:54 From: Doctor Jerry Subject: Re: Relation between a sphere's surface area and its volume Hi Matt, Since S = 4*pi*r^2 and V = (4/3)*pi*r^3, the volume goes up as the cube of the radius and the surface area as the square of the radius. Another way of stating this is to calculate, as you have done, the ratio of the volume to the area. Some numerical illustrations would be useful: Make a small table: r 1 2 3 4 5 V 4.19 33.51 113.10 268.08 523.60 S 12.57 50.27 113.10 201.06 314.16 V/S 0.33 0.67 1 1.33 1.67 Do these values look familiar? I'm going to introduce you to another relation, one you usually wouldn't see until you get to calculus, so don't worry if you don't quite understand what's going on. It will involve some large calculations that would be best done by computer - it would take a long time to do by hand. You might not notice this relation with small values of r, but if you start with a large value of r, and then watch how the volume changes (that is, compute V(r+1) - V(r)), you should get some familiar numbers. Here is a table with r, S(r), and V(r+1) - V(r) for r from 10,000 to 10,010: r S(r) V(r+1) - V(r) 10000 1.25664 * 10^9 1.25676 * 10^9 10001 1.25689 * 10^9 1.25701 * 10^9 10002 1.25714 * 10^9 1.25727 * 10^9 10003 1.25739 * 10^9 1.25752 * 10^9 10004 1.25764 * 10^9 1.25777 * 10^9 10005 1.25789 * 10^9 1.25802 * 10^9 10006 1.25815 * 10^9 1.25827 * 10^9 10007 1.25840 * 10^9 1.25852 * 10^9 10008 1.25865 * 10^9 1.25877 * 10^9 10009 1.25890 * 10^9 1.25903 * 10^9 10010 1.25915 * 10^9 1.25928 * 10^9 For another way to look at this, compute (V(r+1) - V(r))/S(r) for large values of r. You'll notice that this value gets closer to 1. This means that as the radius increases, the volume increases by exactly the surface area value. As I said before, you're probably just looking for the first relation. Consider the second one just as an introduction to some math you'll see in the future. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ |
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