you need to get a volume formula in terms of a single variable.
take the time derivative of the above volume function ... sub in your given values for and ... calculate the value of .
So, my textbook only covers implicit diff. as it relates to ellipses and circles. Needless to say, it all went over my head. Now I have a related rates question where I absolutely need to use implicit diff. Here goes:
A water tank has the shape of an inverted cone with a base radius of 4 meteres and a height of 6 meteres. If water is being pumped into the tank at a rate of 3 m^3/min, find the rate at which the water level is rising when the tank is 4 meters deep. Specify the units of your answer.
So I have two triangles, one with measurements given, and one with only height (because radius at 4m deep is currently unknown)
I also have the formula V = 1/3 pi*r^2*h
Now, to set up a proportion to find r?
-- = --
6r = 4h, in terms of r
r = 4h/6, substitute the known h
r = 8/3m
If I did everything right, I now know the radius of the cone when height = 4m. However, I'm sure I need to implicitly differentiate the volume formula and I'm not sure how.
Since the volume of a cone is , we now see that
Now [implicitly] differentiate both sides with respect to t:
We can now plug in all the known values and solve for .
Can you try to take it from here?