Thomas Calculus 13th [Solutions]

2 3
Section 4.6 Applied Optimization 295
23. The fixed volume is V r h
2
r h V 2r
3 2 3 , where h is the height of the cylinder and r is the radius
r
of the hemisphere. To minimize the cost we must minimize surface area of the cylinder added to twice the
surface area of the hemisphere. Thus, we minimize C 2 2 2 2 8 2
2 rh 4 r 2 r V r 4 V .
2
r 3 r r 3
r
3
Then dC 2V 16
r 0 V
8
r r
3V
. From the volume equation, h V
dr 2
r 3 3 8
2
r
1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3
2
4V 2 3 V 3 2 4 V 2 3 V 3V . Since d C 4V
16
1/3 2/3 1/3 1/3
2 3
3 3 2 3 2
dr r 3
the cost.
1/3
2r
3
0, these dimensions do minimize
24. The volume of the trough is maximized when the area of the cross section is maximized. From the diagram
2 2
the area of the cross section is A ( ) cos sin cos , 0 . Then A ( ) sin cos sin
2
(2sin sin 1) (2sin 1)(sin 1) so A ( ) 0 sin
1
or sin 1 because
sin 1 when 0 . Also, A ( ) 0 for 0 and A ( ) 0 for
2
6
6 2
. Therefore, at
is a maximum.
2
2
6
6 there
25. (a) From the diagram we have: AP x,
RA L x PB 8.5 x,
2
2 2
CH DR 11 RA 11 L x , QB x (8.5 x) ,
HQ 11 CH QB
2 2 2
L x x (8.5 x) ,
2 ,
2 2 2
11 11 L x x (8.5 x)
2 2 2
RQ RH HQ
2 2 2 2
2 2 2
(8.5) L x x (8.5 x ) . It follows that RP PQ RQ
2
2 2 2 2 2 2 2
L x L x x ( x 8.5) (8.5)
2 2 2 2 2 2 2 2 2
L x L x 2 L x 17 x (8.5) 17 x (8.5) (8.5)
2 2 2 2 2 2 2 2 2
17 x 4( L x )(17 x (8.5) ) L x
17 x
2
4[17 x (8.5) ]
3 3 3 3
17x 17x 4x 2x
2
17
2
17 x (8.5) 17x
2
3
( )
4x
4 17
4x
17 2x
8.5
2
.
2
4 x (8x
51)
(b) If f x is minimized, then L is minimized. Now f ( x) f ( x ) 0 when x 51 and
x
2
(4x
17)
8
f ( x ) 0 when x 51
8 . Thus 2
L is minimized when x 51
8 .
(c) When x 51
8 , then L 11.0 in.
2
26. (a) From the figure in the text we have P 2x 2 y y P x . If P 36, then y 18 x . When the
2
cylinder is formed, x 2 r r x and h y h 18 x . The volume of the cylinder is
2
2
2 3
( ) 18x
x
3 x(12 x)
V r h V x . Solving V ( x) 0 x 0 or 12; but when x 0 there is no
4
4
cylinder. Then V ( x) 3 3 x V (12) 0 there is a maximum at x 12. The values of x 12 cm
2
and y 6 cm give the largest volume.
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