A child loves to watch as you refill a transparent plasticbottle with shampoo. E

Question

A child loves to watch as you refill a transparent plasticbottle with shampoo. Every horizontal cross-section of the bottleis a circle, but the diameters of the circles all have differentvalues, so that the bottle is much wider in some places than inothers. You pour in bright green shampoo with constant volume flowrate 16.0 cm3/s. At what rate is its level in the bottle rising at a pointwhere the diameter of the bottle is 6.30cm and at 1.35 cm? A child loves to watch as you refill a transparent plasticbottle with shampoo. Every horizontal cross-section of the bottleis a circle, but the diameters of the circles all have differentvalues, so that the bottle is much wider in some places than inothers. You pour in bright green shampoo with constant volume flowrate 16.0 cm3/s. At what rate is its level in the bottle rising at a pointwhere the diameter of the bottle is 6.30cm and at 1.35 cm?

Explanation / Answer

The shampoo is poured with constant volume flow rate 16.0cm3/s = 16.0 * (10-2)3m3/s = 16.0 * 10-6 m3/s The diameter of the bottle at a point is 6.30 cm and 1.35 cmat the other point. According to the continuity equation,we have A1 * v1 = A2 *v2 Here,v1= 16.0 *10-6m3/s,d1= 6.30 cm = 6.30 *10-2 m and d2= 1.35 cm = 1.35 *10-2 m or ( * (d1/2)2) * v1 =( * (d2/2)2) * v2 or v2 = (d1/d2)2 *v1 = (6.30 * 10-2/1.35 * 10-2) *16.0 * 10-6 or v2 = 74.67 * 10-6 m3/s =74.67 cm3/s Therefore,the rate at which the level in the bottle risingwhere the diameter of the bottle is 1.35 cm is 74.67cm3/s. or v2 = 74.67 * 10-6 m3/s =74.67 cm3/s Therefore,the rate at which the level in the bottle risingwhere the diameter of the bottle is 1.35 cm is 74.67cm3/s.

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