netquiddler: (Default)
[personal profile] netquiddler
Yesterday, in calculus III class, we learned that the formula of a sphere is given by:


where the center is at [;(h,k,l);] and the radius is [;r;]. Now, my professor has decided to give us problems like this:


asking whether this is a formula for a sphere, and if so, what is its center and radius. This problem would be solved like this.





This shows a sphere with a radius of [;2\sqrt{2};] and a center of [;(-2,3,-1);]. (We would not have a sphere if we ended up with a nonpositive radius.)

Now, rather than having to trudge through that work every time we do this problem, I think I'm going to be clever.

Suppose we have a problem in this form:


We can do the same thing to this problem.





This leaves us a sphere if and only if [;\frac{A^2+B^2+C^2}{4}>D;]. Now if [;D;] is negative, then this will always be true. If [;D;] is zero, then [;A;], [;B;], and [;C;] would all have to be zero for this to be false.

Any sphere, then, has a center of [;\left(-\frac{A}{2},-\frac{B}{2},-\frac{C}{2}\right);] and a radius of [;\frac{\sqrt{A^2+B^2+C^2-4D}}{2};].

I think this is a valid shortcut. What say the rest of you?

[To see formulas:]

Date: 2010-09-10 12:35 am (UTC)
From: [identity profile]
I could answer you if my brain did not absolutely abhor how that looks when typed out.

Date: 2010-09-10 01:54 am (UTC)
From: [identity profile]
I think your link's broken, is what I think.

Date: 2010-09-10 02:01 am (UTC)
From: [identity profile]
Also, now that I got it working, it may just be me, but there are ABOMINABLE white borders on all the formulas. Is it doing that for you? @.@

Date: 2010-09-10 03:55 am (UTC)


netquiddler: (Default)

February 2011

2021222324 2526

Most Popular Tags

Style Credit

Expand Cut Tags

No cut tags
Page generated Sep. 20th, 2017 04:01 am
Powered by Dreamwidth Studios