Calculus Funnel Question

Question

A funnel has a radius of 4in and a height of 12in. If sand is leaving the funnel at a rate of

π /2 in3/s, then how fast is the height of the sand in the funnel falling when it is 7 in high?

(Hints: Volume of a cone is given by V =

1 /3 πr2

With stated variables and draw and label diagram and write equation then solve

Answer 1

Let h = height of sand at time t

r = radius of surface of sand at time t

V = volume of sand at time t

Given: dV/dt = -pi/2

Find: dh/dt when h = 7

V = (1/3)(pi)(r^2)h

Using similar triangles, r / h = 4/12. So, r = (1/3)h

V = (1/3)(pi)(1/9)h^3

V = (1/27)(pi)(h^3)

So, dV/dt = (1/9)(pi)(h^2)(dh/dt)

-(pi)/2 = (49/9)(pi)(dh/dt)

dh/dt = -9/98 in/sec

The height is falling at the rate of 9/98 in / sec.