(a) Show that the volume of a segment of height h of a sphere of radius r is

(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 3x3 €“ 9x + 2 =0 where 0 (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 x3 €“ 3rx2 + 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 in3/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?

Transcribed Image Text:

V=nh*(3r - h)