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# rate of something watch

1. How do I find the rate of the object filling? I know you have to use the chain rule, but whats the method?
edit: found a question:
volume of sphere is increasing at a rate of 75cm^3s^-1. Find the rate at which the radius is increasing at the instant when the radius of the sphere is 15cm.
vol of sphere = v=4/3 . Pi . r^3
2. (Original post by SolidSnake)
How do I find the rate of the object filling? I know you have to use the chain rule, but whats the method?
edit: found a question:
volume of sphere is increasing at a rate of 75cm^3s^-1. Find the rate at which the radius is increasing at the instant when the radius of the sphere is 15cm.
vol of sphere = v=4/3 . Pi . r^3
You are told that "volume of sphere is increasing at a rate of 75cm^3s^-1.", which is a time rate of change of the volume, so you can write this down as,

dV/dt - how the volume changes wrt time.

Using the chain rule, we can also write,

dV/dt = dV/dr*dr/dt ----------------------(1)

from V = (4/3)pi.r^3

dV/dr = 4pi.r²

substituting for dV/dr = 4pi.r² and dV/dt = 75 in (1), we get

75 = 4pi.r²*dr/dt

at r=15,

75 = 4pi.225*dr/dt
dr/dt = 1/(12pi) - where dr/dt is the time rate of change of the radius.
============
3. (Original post by SolidSnake)
How do I find the rate of the object filling? I know you have to use the chain rule, but whats the method?

volume of sphere is increasing at a rate of 75cm^3s^-1. Find the rate at which the radius is increasing at the instant when the radius of the sphere is 15cm.
vol of sphere = v=4/3 . Pi . r^3
volume is increasing with respect to time... therefore, dv/dt = 75
you want the rate which the radius is increasing with time... which is dr/dt

volume of a sphere,v, is (4/3)Pi(r^3)
rate volume increases with radius is dv/dr

v=(4/3)Pi(r^3)
dv/dr = 4Pi(r^2)

now, we know dv/dt, dv/dr and want to work out dr/dt

from the chain rule...

dv/dt = dv/dr.dr/dt

therefore...

dr/dt = {dv/dt}/{dv/dr}
dr/dt = 75/[4pi(r^2)]
4. and when the radius is 15cm...
dr/dt = 75/[4pi(225)]
dr/dt = 75/900pi
dr/dt = pi/12 cm.s^-1

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