Page:Calculus Made Easy.pdf/65

of the bottom is ${\displaystyle 200}$ feet. Find an expression for the quantity pouring in or out when the depth of water varies by ${\displaystyle 1}$ foot; hence find, in gallons, the quantity withdrawn hourly when the depth is reduced from ${\displaystyle 14}$ to ${\displaystyle 10}$ feet in ${\displaystyle 24}$ hours.

The volume of a frustum of pyramid of height ${\displaystyle H}$, and of bases ${\displaystyle A}$ and ${\displaystyle a}$, is ${\displaystyle V={\dfrac {H}{3}}(A+a+{\sqrt {Aa}})}$. It is easily seen that, the slope being ${\displaystyle 45^{\circ }}$, if the depth be ${\displaystyle h}$, the length of the side of the square surface of the water is ${\displaystyle 200+2h}$ feet, so that the volume of water is

${\displaystyle {\dfrac {dV}{dh}}=40,000+800h+4h^{2}=}$ cubic feet per foot of depth variation. The mean level from ${\displaystyle 14}$ to ${\displaystyle 10}$ feet is ${\displaystyle 12}$ feet, when ${\displaystyle h=12}$, ${\displaystyle {\dfrac {dV}{dh}}=50,176}$ cubic feet.

Gallons per hour corresponding to a change of depth of ${\displaystyle 4}$ ft. in ${\displaystyle 24}$ hours ${\displaystyle {}={\dfrac {4\times 50,176\times 6.25}{24}}=52,267}$ gallons.

(13) The absolute pressure, in atmospheres, ${\displaystyle P}$, of saturated steam at the temperature ${\displaystyle t^{\circ }}$ C. is given by Dulong as being ${\displaystyle P=\left({\dfrac {40+t}{140}}\right)^{5}}$ as long as ${\displaystyle t}$ is above 80°. Find the rate of variation of the pressure with the temperature at ${\displaystyle 100^{\circ }}$ C.