Activity 3.1.2.
A water tank has the shape of an inverted circular cone (point down) with a base of radius 6 feet and a depth of 8 feet. Suppose that water is being pumped into the tank at a constant instantaneous rate of 4 cubic feet per minute.
(a)
Draw 2-3 pictures of the conical tank that show sketches of the water level at different points in time when the tank is not yet full. Introduce variables that measure the radius of the water’s surface and the water’s depth in the tank, and label them on your figure.
(b)
Say that \(r\) is the radius and \(h\) the depth of the water at a given time, \(t\text{.}\) What equation relates the radius and height of the water, and why?
(c)
Determine an equation that relates the volume of water in the tank at time \(t\) to the depth \(h\) of the water at that time.
(d)
Through differentiation, find an equation that relates the instantaneous rate of change of water volume with respect to time to the instantaneous rate of change of water depth at time \(t\text{.}\)
(e)
Find the instantaneous rate at which the water level is rising when the water in the tank is 3 feet deep.
(f)
When is the water rising most rapidly: at \(h = 3\text{,}\) \(h = 4\text{,}\) or \(h = 5\text{?}\) Why?
