it is, however, always worth while to try whether the expression can be put in a simpler form.
First we must try to bring it into the form some expression involving only.
The expression may be written
.
Squaring, we get
,
which simplifies to
;
or
,
that is
;
hence and .
(4) The volume of a cylinder of radius and height is given by the formula . Find the rate of variation of volume with the radius when in. and in. If , find the dimensions of the cylinder so that a change of in. in radius causes a change of cub. in. in the volume.
The rate of variation of with regard to is
.
If in. and in. this becomes . It means that a change of radius of inch will cause a change of volume of cub. inch. This can be easily verified, for the volumes with and