Question

Sand falls from a hopper at a rate of 0.2 cubic meters per hour and forms a conical pile beneath. Suppose the radius of the cone is always half the height of thecone

a) Find the rate at which the radius of the cone increases when the radius is 2 meters

b) Find the rate at which the height of the cone increases when the radius is 2 meters.

Answer 1

volume of con e= 1/3πr^2 h =

dV/dt = 0.2 m^3 /hr

r = h/2

h = 2r

V = 1/3πr^2 ( 2r) ----------in terms of radius only

V = 2/3πr^3

dv/dt =2/3π ( 3r^2) ( dr/dt)

dv/dt = 2π ( r^2) ( dr/dt)

0.2 = 2( 3.14) ( 4) ( dr/dt)

dr/dt = 0.008 m /hr apprx ( increase in radius)

V = 1/3π( h/2)^2 h ---------in term,s of heght only

V = 1/12π h^3

dv/dt = 3/12πh^2 ( dh/dt)

0.2 = 1/4 ( 3.14) ( 4)^2 ( dh/dt) ( height is twice of radius)

dh/dt = 0.0159 m /hr apprx ( incresae in height)