Application #3: Depth and Volume in a Funnel
DIRECTIONS
Write a piecewise-defined function to represent the situation described, using the format If(interval,expression) for each piece. Add solid or hollow "points" to the graph to clarify what happens when the function rule changes.
THE SITUATION
The funnel in the 3D graph below consists of a cylindrical portion and a conical portion. The radius of the cylinder is 0.5 cm and its height is 2 cm. The truncated cone opens out to a radius of 3 cm and also has a height of 2 cm.
Suppose you close off the bottom (small end) of the funnel when it's held vertically and start adding water. How does the volume of water in the funnel depend on the depth?
Let represent the depth of water in centimeters and represent the volume of water in cubic centimeters.
THE SITUATION, continued
We could also ask the inverse question....
Suppose you close off the bottom (small end) of the funnel when it's held vertically and start adding water. How does the depth of water in the funnel depend on the volume of water you have added??
Let represent the volume of water added in cubic centimeters and represent the depth in centimeters.