Calculus: Analytic Geometry and Calculus, with Vectors

190 Functions, limits, derivatives

When two of the three numbers in this formula are known, we can calcu-
late the remaining one. Some more or less instructive problems involv-
ing such motions require use of the formula

(3.871) lol= C

2 +()2

for the speed of the particle. When motions in E3 are to be investigated,

it may be advantageous to use the vector formulas

(3.88) r=xi+yj+zk

(3.881) v=dti+dtj+atk

in which x, y, z represent the coordinates of the particle at time t. Of

course, this motion reduces to motion in the xy plane when z = 0 for

each t.

Problems 3.89

1 As in Figure 3.891, a rope 13 feet long extends from a boat to a point on

a dock 5 feet higher. A man on the dock is pulling rope in at the rate of 72 feet

Figure 3.891

per minute. How fast is the boat moving?

4ns.: 78 feet per minute.

2 A light atop a pole is H feet above a level

street. A man h feet tall walks steadily, F feet per

second, along a line leading away from the base of

the pole. At what rate is the tip of his shadow

moving when he is x feet from the pole?

_ins.: H F h feet per second.

We may be short on information about formation of raindrops in clouds,

but we can study the growth of a spherical drop during the part of its develop-

ment when, for some constant k, the rate, in cubic centimeters per second,

at which it collects water is the product of i and the area of its surface. At

what rate does the radius increase? fins.: k centimeters per second.

(^4) It is observed that the radii of volatile mothballs decrease at the rate of
0.5 centimeter per year. Find the rate at which mothballstuff is evaporating
from a collection of 100 mothballs of radius 0.6 centimeter. .,lns.: About 226
cubic centimeters per year or about 0.62 cubic centimeter per day.
5 Sand is falling at the rate of 2 cubic feet per minute upon the tip of a
conical sandpile which maintains the form of a right circular cone the height of
which is always equal to the radius of the base. Sketch a figure and calculate the
rate at which the height is increasing when the height is 6 feet. llns.: 1/18vr feet
per minute.
5 Thread is being unwound at the rate of .4 centimeters per second from
an ordinary circular cylindrical spool of radius R centimeters. The unwound
part of the thread has length s and is stretched into a line segment TE tangent