Question

Water is leaking out the bottom of a hemispherical tank of radius 9 feet at a rate of 2 cubic feet per hour. the tank was full at a certain time. how fast is the water level changing when its height h is 8 ​feet? note​: the volume of a segment of height h in a hemisphere of radius r is pi h squared left bracket r minus left parenthesis h divided by 3 right parenthesis right bracket.

Answer 1

The radius  of the hemisphere is:
V=πh²(r-h/3)
because the volume of the hemisphere doesn't change , we measure the change in the volume in relation to the change in height:
dv/dt=π(h²-(1/3 dh/dt))+2h(r-h/3)dh/dt)
dv/dt=π(-h²/3+2hr-2h²/3)dh/dt
dv/dt=π(2hr-h²)dh/dtdv/dt=-2 ft³/h, height is 8 ft and radius of the hemisphere is 9 ft
-2=π(2h*9-h²)dh/dt
-2=π(2*8*9-8²)dh/dt
-2=π(80)dh/dt
dh/dt=-2/(π*80)
dh/dt=-0.008