Page:Calculus Made Easy.pdf/51

This page has been proofread, but needs to be validated.

WHAT TO DO WITH CONSTANTS

31

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 $y=$ some expression involving $x$ only.

The expression may be written

$(a-b)y+(a+b)x=(x+y){\sqrt {a^{2}-b^{2}}}$ .

Squaring, we get

$(a-b)^{2}y^{2}+(a+b)^{2}x^{2}+2(a+b)(a-b)xy$

$=(x^{2}+y^{2}+2xy)(a^{2}-b^{2})$ ,

which simplifies to

$(a-b)^{2}y^{2}+(a+b)^{2}x^{2}=x^{2}(a^{2}-b^{2})+y^{2}(a^{2}-b^{2})$ ;

or

$\left[(a-b)^{2}-(a^{2}-b^{2})\right]y^{2}=\left[(a^{2}-b^{2})-(a+b)^{2}\right]x^{2}$ ,

that is

$2b(b-a)y^{2}=-2b(b+a)x^{2}$ ;

hence $y={\sqrt {\frac {a+b}{a-b}}}x$ and ${\frac {dy}{dx}}={\sqrt {\frac {a+b}{a-b}}}$ .

(4) The volume of a cylinder of radius $r$ and height $h$ is given by the formula $V=\pi r^{2}h$ . Find the rate of variation of volume with the radius when $r=5.5$ in. and $h=20$ in. If $r=h$ , find the dimensions of the cylinder so that a change of $1$ in. in radius causes a change of $400$ cub. in. in the volume.

The rate of variation of $V$ with regard to $r$ is

${\frac {dV}{dr}}=2\pi rh$ .

If $r=5.5$ in. and $h=20$ in. this becomes $690.8$ . It means that a change of radius of $1$ inch will cause a change of volume of $690.8$ cub. inch. This can be easily verified, for the volumes with $r=5$ and $r=6$