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

220 Chapter 2 Polynomial and Rational Functions
77. (a) A xy and
(c)
78. A xy and
x 3y 2 64
Thus,
x 4y 2 30
200
4
0
y 2 64
x 3
y 2 64 2x 58
x 3 x 3
2x 58
A xy x x 3
2xx 29
, x > 3.
x 3
By graphing the area function, we see that A is minimum when x 12.8 inches and y 8.5 inches.
79. (a) Let time from Akron to Columbus and
from Columbus back to Akron.
time
xt1 100 ⇒ t1 100
t1 t2
(b) Vertical asymptote: x 25
Horizontal asymptote: y 25
x
(c) 200
yt 2 100 ⇒ t 2 100
y
50t 1 t 2 200
100
x
y 2 30
x 4
The area is minimum when x 11.75 inches and y 5.87 inches.
The area is minimum when x is approximately 12.
t 1 t 2 4
100
y
4
100y 100x 4xy
25y 25x xy
25x xy 25y
25x yx 25
Thus, y 25x
x 25 .
y 2 30 2x 22
x 4 x 4
2x 22 2xx 11
Thus, A xy x x 4 .
x 4
(b) Domain: Since the margins on the left and right are each 2 inches, x > 4. In interval notation, the domain is 4, .
40
x
5 6 7 8 9 10 11 12 13 14 15
y1 (Area) 160 102 84 76 72 70 69.143 69 69.333 70 70.909
200
3
0
(d)
39
25 65
0
x 30 35 40 45 50 55 60
y 150 87.5 66.67 56.25 50 45.83 42.86
(e) Yes. You would expect the average speed for the round
trip to be the average of the average speeds for the two
parts of the trip.
(f) No. At 20 miles per hour you would use more time in
one direction than is required for the round trip at an
average speed of 50 miles per hour.