Thomas Calculus 13th [Solutions]

190 Chapter 3 Derivatives
35.
1 dy 1 dQ 2
y QD D QD dD 1 (0) 233 ( 2) 466 L/min increasing about 0.2772 L/min
dt dt dt 41 2
(41) 1681
2
36. Let P( x, y ) represent a point on the curve y x and the angle of inclination of a line containing P and the
y
2
origin. Consequently, tan tan x
2
x sec d dx d 2
cos dx . Since dx 10 m/sec and
x
x
dt dt dt
dt dt
2
2 2
cos
x 3 1 , we have d 1 rad/sec.
x 3 2 2 2 2
y x 9 3 10 dt x 3
2 2
37. The distance from the origin is s x y and we wish to find
ds 1 2 2 1/2
(5)( 1) (12)( 5)
2 ( ) 2 2 dy
x y x dx y
dt 5m/sec
(5,12) dt dt
(5,12) 25 144
38. Let s distance of the car from the foot of perpendicular in the textbook diagram tan s
132
sec 2 d 1 ds d cos 2 ds
dt 132 dt dt 132 dt
; ds 264 and 0 d 2 rad/sec. A half second later the car has
dt
dt
2
traveled 132 ft right of the perpendicular | | , cos 1 , and ds 264 (since s increases)
4
2 dt
1
d
( )
2
(264) 1 rad/sec.
dt 132
2
39. Let s 16t represent the distance the ball has
fallen, h the distance between the ball and the
ground, and I the distance between the shadow and
the point directly beneath the ball. Accordingly,
s h 50 and since the triangle LOQ and triangle
2
PRQ are similar we have I 30h
h 50 16t
50 h
2
30(50 16 )
and I t 1500 30 dI 1500
2
50 (50 16 t ) 16t
2 dt 8t
3
dI 1500 ft/sec.
dt t
1
2
40. When x represents the length of the shadow, then tan
We are given that d 0.27 3 rad/ min. At
dt
2000
3 ft/min 0.589 ft/min 7.1in./min.
16
80 2
sec d 80 dx dx
x dt 2
x dt dt
x 60, cos 3 dx
5 dt
x
2 2
sec
80
d
x
2 2
sec
80
d
dt
d 3 5
dt and sec
dt 2000 3
3 3
41. The volume of the ice is 4 4 4 dV 2
V r
4
3 3 dt
r dr dr
dt dt r 6
5 in./min when dV 3
10 in /min,
72
dt
the thickness of the ice is decreasing at 5
2
in/min. The surface area is S 4 r ds 8 r dr dS
72
dt dt dt r 6
48 5 10 2
72 3 in /min, the outer surface area of the ice is decreasing at 10 2
in /min.
3
42. Let s represent the horizontal distance between the car and plane while r is the line-of-sight distance between
2 2
the car and plane 9 s r ds r dr ds 5 ( 160) 200 mph speed of plane speed
dt 2
r 9
dt dt r 5 16
of car 200 mph the speed of the car is 80 mph.
.
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