Calculus question

Question

You decide to warm your house by lighting your wood fire place. The smoke is coming out the chimney in the shape of an inverted right circular cone, the diameter of which is equal to its height. If the rate of smoke emission is 100m^3/min, how fast is the height increasing when the height is 30m?

Help!?

Answer 1

Let V = volume (in cubic meters)of the cone at time t minutes

     h = height (in meters) of the cone at time t minutes

     r = radius (in meters) of the base of the cone at time t minutes

 

Given:  dV/dt = 100 m³/min and h = 2r

 

Find:  dh/dt when h = 30 m

 

Solution:  V = (1/3)πr²h = (1/3)π(h/2)²h

 

                                    = (1/12)πh³

 

             dV/dt = (1/4)πh²(dh/dt)

 

              100 = (1/4)π(30)²(dh/dt)

 

            dh/dt = 4/(9π) m/min