Problem 5P

a Show that the volume of a segment of height h of a sphere of radius r is V=13h2(3rh) See the figure. b Show that if a sphere of radius 1 is sliced by a plane at a distance x from the center in such a way that the volume of one segment is twice the volume of the other, then x is a solution of the equation 3x39x+2=0 where 0x1. Use Newtons method to find x accurate to four decimal places. c Using the formula for the volume of a segment of a sphere, it can be shown that the depth x to which a floating sphere of radius r sinks in water is a root of the equation x33rx2+4r3s=0 where s is the specific gravity of the sphere. Suppose a wooden sphere of radius 0.5 m has specific gravity 0.75. Calculate, to four-decimal-place accuracy, the depth to which the sphere will sink. d A hemispherical bowl has radius 5 inches and water is running into the bowl at the rate of 0.2 in 3/s. i How fast is the water level in the bowl rising at the instant the water is 3 inches deep? ii At a certain instant, the water is 4 inches deep. How long will it take to fill the bowl?

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