Related rate question

Question

Sand is falling off and piling in a conical shape at a rate of 5 ft³/s. The diameter of the pile is equal to the height of the pile. At what rate the height of the pile is changing when the height is 2 ft.

 

Answer 1

Draw the cone with its apex at the top; draw the height & call it x.  Draw the radius of the circular base, labeling it x/2 since the diameter = height of pile.

 

V = (pi/3)(x/2)²(x) using the usual formula "pi over 3 times radius squared times height"

 

V = pi/12(x³) and dV/dt = pi/4(x²)•dx/dt

 

Now's the time to substitute: dV/dt = 5 & x = 2; solve for dx/dt.

 

I believe that the answer is 5/pi ft/sec = 1.59 ft/sec.