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MAXIMA AND MINIMA

109

keep always in the same proportion to each other; that is, at any instant, the cylinder is similar to the original cylinder. When the radius of the base is $r$ feet, the surface area is increasing at the rate of $20$ square inches per second; at what rate is its volume then increasing?

${\begin{aligned}{\text{Area}}&=S=2(\pi r^{2})+2\pi r\times 2r=6\pi r^{2}.\\{\text{Volume}}&=V=\pi r^{2}\times 2r=2\pi r^{3}.\\{\frac {dS}{dr}}&=12\pi r,\quad {\frac {dV}{dr}}=6\pi r^{2},\\dS&=12\pi r\,dr=20,\quad dr={\frac {20}{12\pi r}},\\dV&=6\pi r^{2}\,dr=6\pi r^{2}\times {\frac {20}{12\pi r}}=10r.\end{aligned}}$

The volume changes at the rate of $10r$ cubic inches.

Make other examples for yourself. There are few subjects which offer such a wealth for interesting examples.

Exercises IX. (See page 258 for Answers.)

(1) What values of $x$ will make $y$ a maximum and a minimum, if $y={\dfrac {x^{2}}{x+1}}$ ?

(2) What value of $x$ will make $y$ a maximum in the equation $y={\dfrac {x}{a^{2}+x^{2}}}$ ?

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