Question

Sand is pouring from a pipe at the rate of 12 cm/s. The falling sand forms a cone
on the ground in such a way that 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?​

Answer 1

Answer:

Let r=radius;h=height;V= Volume of sand cone, t=time

Given  h=4 cm;  dV/ dt = 12 cm³/s and h= 1 / 6  r (or) r=6h

∴V=  1 /3 πr² h

=   1 /3  π(6h)²   h = ( 36h³ π )/3 = 12πh³   (diff w.r.t. t)

∴dV / dt  = 12π.3h²  dh  / dt

12=12π.3(4)  ²dh /dt

 

∴   dh /dt  = 1 /48π  cm/s.

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