Thomas Calculus 13th [Solutions]

Section 4.6 Applied Optimization 299
(b)
(c)
Both graphs indicate the same maximum value and are consistent with each other. Changing k does not
change the dimensions that give the strongest beam (i.e., do not change the values of w and d that produce
the strongest beam).
2 2
3 3 2 1/2
44. (a) From the situation we have w 144 d . The stiffness of the beam is S kwd kd (144 d ) ,
2 2
4 kd (108 d )
where 0 d 12. Also, S ( d )
critical points at 0, 12, and 6 3. Both d 0 and d 12
2
144 d
cause S 0. The maximum occurs at d 6 3. The dimensions are 6 by 6 3 inches.
(b)
(c)
Both graphs indicate the same maximum value and are consistent with each other. The changing of k has
no effect.
45. (a) s 10cos( t) v 10 sin( t ) speed |10 sin( t)| 10 |sin( t ) | the maximum speed is
10 31.42 cm/sec since the maximum value of |sin ( t )| is 1; the cart is moving the fastest at t 0.5 sec,
1.5 sec, 2.5 sec and 3.5 sec when |sin ( t )| is 1. At these times the distance is s 10cos 0 cm and
2
2 2 2
a 10 cos ( t) | a| 10 |cos ( t) | | a| 0 cm/sec
2
(b) | a| 10 |cos ( t )| is greatest at t 0.0 sec, 1.0 sec, 2.0 sec, 3.0 sec, and 4.0 sec, and at these times the
magnitude of the carts position is | s | 10 cm from the rest position and the speed is 0 cm/sec.
46. (a) 2sin t sin 2t 2sin t 2sin t cos t 0 (2sin t)(1 cos t) 0 t k where k is a positive integer
2
1/2
2 1/2
(b) The vertical distance between the masses is s( t) | s1 s2 | ( s1 s2) ((sin 2t 2sin t) )
1
2 1/2
2(cos 2t 2cos t)(sin 2t 2sin t)
s ( t) ((sin 2t 2sin t) ) (2)(sin 2t 2sin t)(2cos 2t 2cos t)
2 |sin 2t
2sin t|
4(2cos t 1)(cos t 1)(sin t)(cost
1)
critical times at 0, 2 , , 4 , 2 ; then s(0) 0,
|sin 2t
2sin t|
3 3
2 4 2 3 3
s sin 2sin ,
3 3 3 2
the greatest distance is 3 3
2
4 8 4 3 3
s( ) 0, s sin 2sin , s(2 ) 0
3 3 3 2
at t 2 and 4 3 3
47. (a)
(b)
2 2 2 2 1/2
s (12 12 t) (8 t) ((12 12 t) 64 t )
ds 1
2 2 1/2 208 144
2 ((12 12 t ) 64 t ) [2(12 12 t )( 12) 128 t ] t ds 12 knots and
dt
2 2
(12 12 t ) 64t
dt t 0
ds 8 knots
dt t 1
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