If a fluid enters a hemispherical vat with a radius of 10meter . at a rate of 0.5cubic meter per minute, how fast will the fluid be rising when the depth is 5meter?

Answer 1

Trick:

at depth h
the volume increase with delta h
= the surface area * delta h

draw that and remember it.
surface area = pi r^2

what is r of surface at h = 5 ?
that takes some geometry
triangle from center
5^2 + r^2 = 10^2
r^2 = 75
r = 5 sqrt 3
so
dV = pi (75) dh
dV/dt = 75 pi dh/dt
so
dh/dt = .5 ft^3/min / (75 pi ft^2