How fast is the volume of the ice increasing at this time? Round your answer to three decimal places.

Question

Ice is being formed in the shape of a circular cylinder with inner radius 1 cm and height 5 cm. The outer radius of the ice is increasing at 0.06 cm per hour when the outer radius is 1.5 cm. How fast is the volume of the ice increasing at this time?

Round your answer to three decimal places.

Answer 1

 

   V = pi * radius * radius * height = pi * radius^2 * height

 

   pi is approximately 22/7 and height is given to fixed at 5

   BUT radius is a function of time, so the chain rule is required.

   Moreover, when R=1.5 then dR/dt 0.06 as given in the problem

 

   Substituting, V = 110/7* radius^2 = 110/7*R^2

 

   Differentiating

 

          dV/dt = 220/7 * R * dR/dt

 

         dV/dt = 220/7 * 1.5 * 0.06

 

                 = 220/7 * 15/10 * 6/100

 

                 = (220*15*6)/(7*10*100)

 

                 = (220*90)/(7*10*100)

 

                 = (22*9)/70   <-- factor of 100 cancels

 

                 = 198/70

 

                = 2 and 58/70

 

                = 2.82857142857142857142857142857....

 

and that is is cubic centimeters per hour