Rates of Change

Question

A cylindrical jar,of radius 3 cm, contains water to a depth of 5 cm. The water is then poured at a steady rate into an inverted container with its axis vertical. After t seconds, the depth of the water in this container is x cm and the volume, V ml, of water that has been transferred by V=1/3∏x^3.Given all the water is transferred in 3 seconds,find

 

(a) dV/dt in terms of ∏(pi)

(b)the rate at which x is increasing at the moment when x=2.5

 

 

Ans for a) 15∏ cubic cm

           b)2.4cm s^-1

Answer 1

a) volume of water in cylinder = Πr²h = Π(3²)5 = 45Π cm³

 

since it's all poured out in 3 sec, then dV/dt = 45Π/3 = 15Π cm³/s

 

b) V = 1/3Πx³ , so differentiating wrt t, dV/dt = 1/3*3Πx²dx/dt

 

   = Πx²dx/dt     so, substituting:  15Π = Π(2.5)²dx/dt

 

So 15/(2.5)² = dx/dt  = 2.4 cm/s