James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

248 Chapter 3 Differentiation Rules
relates x, y, and z is given by the Pythagorean Theorem:
z 2 − x 2 1 y 2
Differentiating each side with respect to t, we have
2z dz
dt
− 2x
dx
dt
1 2y
dy
dt
dz
dt − 1 Sx dx
z dt dtD 1 y dy
When x − 0.3 mi and y − 0.4 mi, the Pythagorean Theorem gives z − 0.5 mi, so
dz
dt − 1 f0.3s250d 1 0.4s260dg
0.5
− 278 miyh
The cars are approaching each other at a rate of 78 miyh.
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20
¨
FIGURE 5
x
ExamplE 5 A man walks along a straight path at a speed of 4 ftys. A searchlight is
located on the ground 20 ft from the path and is kept focused on the man. At what rate
is the searchlight rotating when the man is 15 ft from the point on the path closest to
the searchlight?
SOLUTION We draw Figure 5 and let x be the distance from the man to the point on the
path closest to the searchlight. We let be the angle between the beam of the searchlight
and the perpendicular to the path.
We are given that dxydt − 4 ftys and are asked to find dydt when x − 15. The
equation that relates x and can be written from Figure 5:
x
− tan x − 20 tan
20
Differentiating each side with respect to t, we get
dx
dt − 20 sec2 d
dt
so
d
dt
− 1 20 cos2 dx
dt
− 1 20 cos2 s4d − 1 5 cos2
When x − 15, the length of the beam is 25, so cos − 4 5 and
d
dt
− 1 5
S 4 5D2
− 16
125 − 0.128
The searchlight is rotating at a rate of 0.128 radys.
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