Chapter 5.P, 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

$3x^3-9x+2=0$

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 $r$ sinks in water is a root of the equation

$x^3-3r{x}^2+4r^{3}s=0$

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?