MathDB

In Euclidean space, take the point

N(0,\ 0,\ 1)

on the sphere

S

with radius 1 centered in the origin. For moving points

P,\ Q

S

such that NP \equal{} NQ and \angle{PNQ} \equal{} \theta \left(0 < \theta < \frac {\pi}{2}\right), consider the solid figure

T

in which the line segment

PQ

can be passed. (1) Show that

z

coordinates of

P,\ Q

are equal. (2) When

P

is on the palne z \equal{} h, express the length of

PQ

in terms of

\theta

and

h

. (3) Draw the outline of the cross section by cutting

T

by the plane z \equal{} h, then express the area in terms of

\theta

and

h

. (4) Pay attention to the range for which

h

can be valued, express the volume

V

T

in terms of

\theta

, then find the maximum

V

when let

\theta

vary.

Problem Statement