Application of Derivatives

Name: Wired Faculty - The Best Learning App
Rating: 4.6 (1864 reviews)

Question

Sand is pouring from a pipe at the rate of 12 cm³/s. The falling sand forms a cone on the ground in such a way what the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand-cone increasing when the height is 4 cm?

Solution

Let r be the radius and h be the height of the right circular cone formed by the falling sand at any time t.

therefore space space space space space straight h space equals space 1 over 6 straight r space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space straight r space equals space 6 space straight h

Let V be the volume of the sand-cone at time t.

therefore space space space space space straight V equals 1 third πr squared straight h space equals space 1 third straight pi left parenthesis 6 straight h right parenthesis squared straight h space equals space 12 space πh cubed therefore space space space space space space dV over dt space equals space 12 straight pi. space space 3 straight h squared space dh over dt space equals space 36 space πh squared space dh over dt space space space space space space

But space space space dV over dt space equals space 12 space cm cubed divided by straight s

(given)

therefore space space space space space 12 space equals space 36 space straight pi space straight h squared space dh over dt space space space space space space space space space space space space space space space space rightwards double arrow space space space space dh over dt space equals space fraction numerator 1 over denominator 3 πh squared end fraction space space space When space straight h space equals space 4 comma space space space space space dh over dt space equals space fraction numerator 1 over denominator 3 straight pi cross times 16 end fraction space equals space fraction numerator 1 over denominator 48 space straight pi end fraction cm divided by straight s

therefore space space space space space the space height space of space the space sand minus cone space is space increasing space at space the space rate space of space fraction numerator 1 over denominator 48 space straight pi end fraction cm divided by straight s space when space its space height space is space 4 space cm.