00:01
So if we have a cone, let's draw a cone here.
00:06
Water is draining from this cone, from a hole cut out at the bottom.
00:12
And this cone is designed so, this water level here.
00:19
The water level in this cone, or the depth of that water, we'll call it the height.
00:26
Then we have the radius of that water level.
00:30
The height is always twice the radius, or two times the radius.
00:35
The radius so we could say h equals two times r if we wanted to find the volume of this amount of water that's in the cone a volume for a cone is volume equals one -third times pi r squared times height and then we know the height is to r so we can substitute that into this equation that would give the volume to be one -third times pi times r squared times the height is 2r and then we can do some simplification here that would give us two -thirds pi r cubed that gives us a formula for the volume okay now when we start talking about the derivative some derivatives we have to think about that the water is coming out of this comb and the radius is going to get smaller as the volume gets smaller.
01:38
So if we define some variables here regarding some derivatives, dv over dt or the derivative of volume with respect to time, this notation represents the rate of change of volume with respect to time.
02:12
And then if we look at the derivative of the radius, the derivative of the radius with respect to time or drdt, that notation represents the rate of change of the radius with respect to time.
02:38
So if we do the derivative of this equation here, if we could implicitly differentiate both sides, this would give us the derivative of volume with respect to time, is we do the power rule here, multiply three times two -thirds pi, that we give us two pi r squared.
02:59
And then we have to multiply by the derivative of r since we're differentiating with respect to time.
03:07
So multiply by the r over dt.
03:11
Okay, so we've done our derivative work...