Thomas Calculus 13th [Solutions]

Section 8.9 Probability 633
36. For parts (a) and (b) the density is
X /16
(1/16) e . For parts (c) and (d) the density is
X /32
(1/ 32) e .
(a)
(b)
(c)
(d)
30 X /16 15/8 5/8
e dX e e
10
25
(1/16) 0.382
X /16 25/16
(1/16) e dX e 0.210
50 X /32 25/16 35/32
e dX e e
35
(1/ 32) 0.125
20 X /32 5/8
e dX e
0
(1/ 32) 1 0.465
37. The expected payout per printer is
expected refund total for 100 machines is $10,256.
1 X /2 2 X /2
0 1
200 (1/ 2) e dX 100 (1/ 2) e dX $102.56. Thus the
2
1 X / c
2/ c 1
2
38. To find the density, solve e dX e 1 , which gives c . The probability of failure
0 c
2
ln(2)
1 X ln 2
ln 2
in the first year is
2
2
e dX 1 0.293. We expect (150)(0.293) 43.934 or about 44 copiers
2 2
0
to fail during the first year.
For Exercises 3952, the density function is
39. 162, 28
193
165
167
148
f ( X ) dX 0.323;
f ( X ) dX 0.262;
about 323 children
about 262 children
40. 20.11, 4.7
1
f ( X ) dX f ( X ) dX 0.74593
17 2 17
41. 55, 4
60
0
f ( X ) dX 0.89435
42. 22,000, 4000
(a) 18,000 18,000
2
1 x
1
f
2
( X ) e with and as given in the solution.
2
1
f ( X ) dX f ( X ) dX 0.84134; (4000)(0.84134) 3365 tires
2
(b) We want to find L such that f ( X ) dX 0.9. A CAS gives 16,874,
L L
lifetime of at least 16,874 miles.
43. 65.5, 2.5
(a)
(b)
68
64
61
1 68
f ( X ) dX f ( X ) dX 0.159, or 16%.
2
f ( X ) dX 0.23832
so 90% of tires will have a
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