What will the rate be?
Please help!
A child loves to watch as you refill a transparent plastic bottle with shampoo. Every horizontal cross-section is a circle, but the diameters of the circles have different values, because the bottle is much wider in some places than in others. You pour in bright green shampoo with constant volume flow rate 15.5 cm3/s.
(a) At what rate is the level of the shampoo in the bottle rising at a point where the diameter of the bottle is 6.50 cm?
(b) At what rate is the level of the shampoo rising at a point where the diameter is 1.25 cm?
I have no idea how to approach this problem
The volume cross-sections are cylinders with infinitestimal height:
dV = πr² dh
dV/dt = π(D/2)² (dh/dt)
(a)
15.5 cm³/s = π(6.50 cm / 2)² (dh/dt)
dh/dt = 0.467 cm/s
(b)
15.5 cm³/s = π(1.25 cm / 2)² (dh/dt)
dh/dt = 12.6 cm/s
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