Question

Sand is pouring from a pipe at the rate of 15 cm³/minute. The falling sand forms a cone on the ground such that the height of the cone is always one-third of the radius of the base. How fast is the height of the sand cone increasing at the instant when the height is 4 cm? 

Answer 1

Given, `(dv)/(dt)` = 15 cm³/min

h = `1/3 r, h = 4`

Since, V = `1/3 pir^2h`

V = `1/3 pi(3h)^2 xx h`

∴ V = `9/3 h^3pi = 3h^3pi`

∴ `(dv)/(dt) = 9h^2pi (dh)/(dt)`

∴ `15 = 9h^2pi (dh)/(dt)`

∴ `15/(9 xx (4)^2 pi) = (dh)/(dt)`

∴ `5/(48pi) = (dh)/dt`