Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/169

22. An elevated car is 40 ft. immediately above a surface car, their tracks intersecting at right angles. If the speed of the elevated car is 16 miles per hour and of the surface car 8 miles per hour, at what rate are the cars separating 5 minutes after they meet? Ans. 17.9 miles per hour.

23. One ship was sailing south at the rate of 6 miles per hour; another east at the rate of 8 miles per hour. At 4 P.M. the second crossed the track of the first where the first was two hours before; (a) how was the distance between the ships changing at 3 P.M.? (b) how at 5 P.M.? (c) when was the distance between them not changing? Ans. (a) Diminishing 2.8 miles per hour; (b) increasing 8.73 miles per hour; (c) 3: 17 P.M.

24. Assuming the volume of the wood in a tree to be proportional to the cube of its diameter, and that the latter increases uniformly year by year when growing, show that the rate of growth when the diameter is 3 ft. is 36 times as great as when the diameter is 6 inches.

25. A railroad train is running 15 miles an hour past a station 800 ft. long, the track having the form of the parabola

$y^{2}=600x,$,

and situated as shown in the figure. If the sun is just rising in the east, find how fast the shadow S of the locomotive L is moving along the wall of the station at the instant it reaches the end of the wall.

Track with shape of a parabola.

Solution.	$y^{2}$	$=600x$
	$2y{\frac {dy}{dt}}$	$=600{\frac {dx}{dt}}$,
or	${\frac {dx}{dt}}$	$={\frac {y}{300}}{\frac {dy}{dt}}$.

Substituting this value of ${\frac {dx}{dt}}$ in

	${\frac {ds}{dt}}$	$={\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}$, we get
(D)	$\left({\frac {ds}{dt}}\right)^{2}$	$=\left({\frac {y}{300}}{\frac {dy}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}$.
Now	${\frac {ds}{dt}}$	= 15 miles per hour dt
		= 22 ft. per sec.
	$y$	= 400 and ${\frac {dy}{dt}}$ = ?

Substituting back in (D), we get

	$\ (22)^{2}$	$=\left({\frac {16}{9}}+1\right)\left({\frac {dy}{dt}}\right)^{2}$,
or,	${\frac {dy}{dt}}$	= $13{\frac {1}{5}}$ ft. per second. Ans.

26. An express train and a balloon start from the same point at the same instant. The former travels 50 miles an hour and the latter rises at the rate of 10 miles an hour. How fast are they separating? Ans. 51 miles an hour.