sand is poured from a conveyer belt onto pile at the rate of 36pi m^3/min. The ratio of the radius to height of the circular conical pile is 2:3. How fast is the height increasing 8 min after pouring began?

Answer 1

radius : height = 2 : 3
r/h = 2/3
r = 2h/3

V = (1/3)π r^2 h
= (1/3)π (4h^2/9)(h)
= (4/27)π h^3
dV/dt = (4/9)π h^2 dh/dt
36π = (4/9)π h^2 dh/dt
81= h^2 dh/dt

after 8 minutes the volume was 8(36π) m^3 = 288π m^3
then (4/27)π h^3 = 288π
h^3= 1944
h = 1944^(1/3) = 12.481

so back to 81= h^2 dh/dt
dh/dt = 81/12.481^2 = appr .52 m/min

Answer 2

thank you I got the same answer😀 but negative 😥cause I still confused if we should take the rate as negative because the question asked how fast is the height increasing?
I think yours is the right
thanks a lot 🙏