Intro Rate of Change with a Rectangular Tank
- Author:
- Tuyetdong Phan-Yamada, David Gurney
Suppose that a rectangular aquarium is being filled with water. The tank is 4 feet long by 2 feet wide by 3 feet high, and the hose that is filling the tank is delivering water at a rate of 0.5 cubic feet per minute.
Use the sliders to change the length and width of the tank, and also to change the inflow rate, i.e. the rate of flow of water into the tank. The length and width are in feet. The inflow rate is in cubic feet per minute.
The key point in doing these problems is to remember that rates
can be thought of as derivatives. In this case, the inflow rate is
the derivative of volume with respect to time, and the rate at which
the height is increasing is the derivative of height with respect to
time.
Also note that the final answer comes from dividing the inflow rate
by the product of the length times the width. Thus, the units are
cubic feet per minute divided by feet squared, which results in
feet per minute - reasonable units for rate of change of the height.