1. A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.
radius, r
height, h
Need to know the volume, V

2. The volume of a cone is given by:
Product Rule
There are 3 variables, V, r, h

3. We want to find the “rate of change of the depth of the water”.
We want dh/dt
This is still the volume of a cone

4. To find dh/dt, we need to know how many things?4
From the problem:
We still need the radius, r, and its rate of change

5. We have to review Geometry and similar triangles:5 12 r
Water level
h = 8
From the problem, we know the radius of the tank is 5 feet, the height of the tank is 12 feet and the water is 8 feet deep. From this we can find the radius of the water in the tank

6. The radius of the water, r, is 10/3 feet.
Now, we need to find dr/dt. Again, we need to use similar triangles.
No matter what the level of the water, we can relate the radius, r, to the depth, h

7. Now, take the derivative of each side with respect to t
This gives the last thing we need since we know dh/dt

8. Make the substitution
Plug in the known numbers and solve for dh/dt

9. Of course, this is left for the student to do.