A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top. if water is poured into the cup at a rate of 2cm3/s, how fast is the water level rising when the water is 5 cm deep?

Accepted Solution

 h: height of the water
 r: radius of the circular top of the water 
 V: the volume of water in the cup.
 We have:
 r/h = 3/10
 r = (3/10)*h
 the volume of a cone is: 
 V = (1/3)*π*r^2*h
 V (t) = (1/3)*π*((3/10)*h(t))^2*h(t)
 V (t) =(3π/100)*h(t)^3
 Using implicit differentiation:
 V'(t) = (9π/100)*h(t)^2*h'(t)
 Clearing h'(t)
 the rate of change of volume is V'(t) = 2 cm3/s when h(t) = 5 cm.
 h'(t) = 8/(9π) cm/s
 the water level is rising at a rate of: 
 h'(t) = 8/(9π) cm/s