A spherical mothball (oh joy)...

L

Lizzie

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A spherical mothball shrinks in such a way that the radius decreases by 1 centimeter per month. How fast is the volume of the mothball changing when the radius of the moth ball is 2 centimeters? (Assume the mothball stays spherical as it shrinks.) [Note: V=(4Pi r<sup>3</sup>)/3

Alrighty, I am just making sure that I'm doing this thing right.

First, I put in 2 cm for r to get V=[(4)(3.14159)(2<sup>3</sup>)]/3?
Which would be simplified to V= 33.150293333333333 and so on.

If I am even on the right track, what do I do from there?

I know that the volume of the mothball at a radius of 2 centimeters is 33.15029333... Now, I need to find how fast the volume is changing at this point. And what do I do with the information about the mothball shrinking 1cm worth of radius/month. I know I'm going to have to make an equation including the 1cm/month thing and the rate at which the volume is changing. Any help?

-Thanks!
-Lizzie
 
FYI: Try never to round until the very end. Round-off error can kill you.

You have V = (4/3)(pi)(r<sup>3</sup>). You are given that dr/dt = -1, and are asked to find dV/dt when r = 2.

When r = 2, the exact value of V is (4/3)(pi)(8) = (32/3)(pi).

To find dV/dt, differentiate "V = (4/3)(pi)(r<sup>3</sup>)" with respect to time "t". Then plug in the known values for r, dr/dr, and V. Solve for dV/dt. (You should get a negative value.)

Eliz.
 
a silly question here but how does 4 times Pi times r cubed all divided by three get simplified to 4/3 times Pi times r cubed? I might have kinda skipped a step in my head :) and thanks for your help once again :)
 
Multiplication is associative.

. . . . .\(\displaystyle \Large{\frac{4\pi r^3}{3}}\)


. . . . .\(\displaystyle \Large{\frac{1}{3}\mbox{ }\left(\frac{4\pi r^3}{1}\right)}\)


. . . . .\(\displaystyle \Large{\frac{1}{3}\mbox{ }\left(\frac{4}{1}\right)\mbox{ }\left(\pi r^3\right)}\)


. . . . .\(\displaystyle \Large{\frac{4}{3}\mbox{ }\pi r^3}\)

Eliz.
 
haha...i knew that :shock: just seeing if you knew...you know, like testing you :lol:
 
Oh yeah! I did it oh yeah! Go stapel! Go stapel! *dancing like an idiot*
 
lol, can you put html on your signature? I tried to put a picture up and it didnt show it, it just showed the html.
 
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