Thomas Calculus 13th [Solutions]

Section 3.10 Related Rates 191
43. Let x represent distance of the player from second base and s the distance to third base. Then dx 16 ft/sec
dt
2 2
(a) s x 8100 2s ds 2 x dx ds x dx . When the player is 30 ft from first base, x 60 s 30 13
and ds
dt
dt dt dt s dt
( 16) 8.875 ft/sec
60 32
30 13 13
1
(b) 90
d
sin 1 cos
90 ds d 1 90 ds 90 ds
s 1
.
dt 2 2
s dt dt s cos dt s k dt
1
d 1 90 32 8
dt 30 13 (60) 13 65 rad/sec; cos 90
d 2
2 sin
s
2 dt
d 1
(c)
dt
90 ds
d 2
. Therefore, x 60 and s 30 13
90 32
s k dt
dt 30 13 (60) 13
90
ds
dt
90
2
cos
2
s 1 s
x
s
x
s
dx
dt
90 1 d
lim ( 15) rad/sec;
2
x 0 x 8100 6
dt
90 dx d 2
lim 1 rad/sec
2
x 8100 dt
x 0
dt 6
2
90 dx
2
s dt
90
2
s sin
2
ds
dt
Therefore, x 60 and s 30 13
90 ds d
2
s dt dt
2
8 rad / sec.
65
90 dx
1
lim d
2
x 8100 dt
x 0
dt
90 x dx 90 dx
s s dt dt
2 x
2
s
s
44. Let a represent the distance between point O and ship A, b the distance between point O and ship B, and D the
2 2 2
distance between the ships. By the Law of Cosines, D a b 2abcos120 dD 1 2a
da 2b
db
dt 2D dt dt
1 2 da 2 db db da .
2D a dt b dt a dt b dt
When a 5, da
dt
14, b 3, and db 21, then dD 413 where D 7. The
dt
dt 2D
ships are moving dD 29.5 knots apart.
dt
90
2
s sin
45. The hour hand moves clockwise from 4 at 30 /hr 0.5 /min. The minute hand, starting at 12, chases
the hour hand at 360 /hr 6 /min. Thus, the angle between them is decreasing and is changing at 0.5 /min
6 /min 5.5 /min.
3 a b
46. The volume of the slick in cubic feet is V where a is the length of the major axis and b is the
4 2 2 ,
length of the minor axis.
3 3
substitute:
dV
dt
dV a d b b d a db da
dt 4 2 dt 2 2 dt 2 16 dt dt
.
3 3 3
3
16
2(5280)(10)
4
(5280)(30)
16
(224,400) 132,183 ft /hr
2
ds
dt
a b Convert all measurements to feet and
Copyright
2014 Pearson Education, Inc.