Page:Calculus Made Easy.pdf/51

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

which simplifies to $(a-b)^2y^2+(a+b)^2x^2=x^2(a^2-b^2)+y^2(a^2-b^2)$; 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 $$V=\pi r^2h$$. 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 $$r=6$$