Chapter 6, Problem 5P

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

$V=\frac{1}{3}\pi h^2\left(3r-h\right)$

(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

3x³ − 9x + 2 = 0

where 0 < x < 1. Use Newton’s 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 s inks 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?

FIGURE FOR PROBLEM 5