C H A P T E R 2 Polynomial and Rational Functions

164 Chapter 2 Polynomial and Rational Functions
89. (a)
90. (a)
(b)
x > 0,
12 x > 0,
x < 12
Domain: 0 < x < 6
212 x 46 xx
8x12 x6 x
6 x > 0
x < 6
(c)
Volume l w h 24 2x24 4xx (c)
91. (a) A l w 12 2xx 2x square inches
2 12x
(b)
Thus, Vx 36 2x36 2xx x36 2x2 .
(b) Domain:
(d)
Volume l w h
height x
length width 36 2x
The length and width must be positive.
3600
0 18
0
The maximum point on the graph occurs at x 6.
This agrees with the maximum found in part (c).
16 feet 192 inches
V l w h
0 < x < 18
12 2xx192
384x cubic inches
2 2304x
(c) Since x and 12 2x cannot be negative, we have
0 < x < 6 inches for the domain.
(d) When the volume is a
maximum with V 3456 in. . The
dimensions of the gutter cross-section
are 3 inches 6 inches 3 inches.
3
x V
x 3,
0 0
1 1920
2 3072
3 3456
4 3072
5 1920
6 0
(e)
The volume is a maximum of 3456 cubic inches when the
height is 6 inches and the length and width are each
24 inches. So the dimensions are 6 24 24 inches.
720
600
480
360
240
120
Box Box Box
Height Width Volume, V
1
2
3
4
5
6
7 736 272 636 26
36 27
3388
2 36 26
3456
V
36 21
36 22
36 23
36 24
36 25
1 2 3 4 5 6
136 21 2 1156
236 22 2 2048
336 23 2 2700
436 24 2 3136
536 25 2 3380
x 2.6 corresponds to a maximum of about
665 cubic inches.
4000
0
0
Maximum: 3, 3456
The maximum value is the same.
6
x
(f) No. The volume is a product of the constant length and
the cross-sectional area. The value of x would remain the
same; only the value of V would change if the length was
changed.