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.

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