Calculus: Analytic Geometry and Calculus, with Vectors

192 Functions, limits, derivatives

10 Let 0 be the angle between the lines of Figure 3.893 that have lengths b
and x. Show that
do acosBdO

at =bcos0d

Hint: At each time the numbers a sin 0 and b sin 0 are equal to each other

because they are both equal to the distance (or the negative of the distance)

from P to the line having length x. Thus we use a slight extension of the trigo-

nometric law of sines.

11 A circle of radius R has its center at the point (0,R) of an xy plane. A

motorcycle is racing at night along the circle in the first quadrant toward the

origin. When the motorcycle is at distance s (measured along the circle) from

the origin, its headlight illuminates a spot at the point (x,0) on the x axis. Show

how the rate at which the spot is moving depends upon R, s, and the rate at

which the motorcycle is moving. Outline of solution: Study of an appropriate

figure can lead us to the first and then the second of the formulas

(1) R= tan q5, x=Rtan2R'

where 0 is the angle between the vectors running from the center of the circle

to the origin and the point (x,0), and

length of arc s/2_ s

(2) radius R

_

2R

Since

(3)

d d sin cos2 ¢ + sin2 4) do_^2 dq

dttan =dt cos 4 - cost 0 dt -

secdt'

differentiating with respect to t gives the answer

(4)

dxIsec s ds

dt - 2 2R dt

Remark: In the context of the motorcycle problem, dr/dt and dx/dt are both

negative. The answer will give very interesting information to those who study

it. Those who do not use the metric system for measuring distances and speeds

may observe that if s and R are numbers of feet and ds/dt is a number of miles

per hour (or furlongs per fortnight), then dx/dt will be a number of miles per hour

(or furlongs per fortnight).

12 A man or a boy or a particle is, for reasons that are sometimes explained,

at the point P1(xi,yi,z,) and is moving with speed q, in the direction of the unit

vector a,i + b,j + c,k. A second animate or inanimate object is at the point

Ps(xs,ys,zs) and is moving or being moved with speed q2 in the direction of the

unit vector aai + bd + c2k. We are required to find the rate at which the dis-

tance between the two objects is changing. Do it. Solution: Let the bodies

beat the points P(x,y,z) and Q(u,v,w) at time t. The distance between the bodies

at time t is then the positive number s for which

(1) s2 = (U ` x)s + (o- Y)s -}- (w - z)2.