A solid spherical shell is centred at the origin

MATHEMATICS · Multi-part · CBSE · Mathematics · Introduction to Three-Dimensional Geometry · Introduction to Three-Dimensional Geometry

Problem

A solid spherical shell is centred at the origin. Its inner radius is

5

units and its outer radius is

7

units. (i) Show that a point

R(x,y,z)

lies in the material of the shell if and only if

25< x^{2}+y^{2}+z^{2}\le 49

. (ii) A point

R

moves along the line whose parametric form is

R(1+2t, 2+3t, 3-t)

, where

t\in\mathbb{R}

. Determine the set of all real values of

t

for which the point

R

is inside the shell. (iii) Hence, find the length of the interval of

t

for which

R

stays within the shell.