Solution: C D 2 r )
r 1 D D 0.159154943. C 2 r D 0.1592 C
To four signiﬁcant digits we have
Problem 1.2 The base of natural logarithms is e D 2.718281828 . . . (a) (b) (c) Express e to ﬁve signiﬁcant digits. Determine the value ofe2 to ﬁve signiﬁcant digits. Use the value of e you obtained in part (a) to determine the value of e2 to ﬁve signiﬁcant digits.
Solution: The value of e is: e D 2.718281828
(a) (b) (c) To ﬁve signiﬁcant ﬁgures e D 2.7183 e2 to ﬁve signiﬁcant ﬁgures is e2 D 7.3891 Using the value from part (a) we ﬁnd e2 D 7.3892 which is not correct in the ﬁfth digit.
[Part (c) demonstrates the hazard ofusing rounded-off values in calculations.]
Problem 1.3 A machinist drills a circular hole in a panel with a nominal radius r D 5 mm. The actual radius of the hole is in the range r D 5 š 0.01 mm. (a) To what number of signiﬁcant digits can you express the radius? (b) To what number of signiﬁcant digits can you express the area of the hole?
a) The radius is in the range r1 D 4.99 mmto r2 D 5.01 mm. These numbers are not equal at the level of three signiﬁcant digits, but they are equal if they are rounded off to two signiﬁcant digits. Two: r D 5.0 mm b) The area of the hole is in the range from A1 D r1 2 D 78.226 m2 to A2 D r2 2 D 78.854 m2 . These numbers are equal only if rounded to one signiﬁcant digit: One: A D 80 mm2
Problem 1.4 The opening in the soccer goal is 24 ftwide and 8 ft high, so its area is 24 ft ð 8 ft D 192 ft2 . What is its area in m2 to three signiﬁcant digits? Solution:
A D 192 ft2 1m 3.281 ft
D 17.8 m2
A D 17.8 m2
Problem 1.5 The Burj Dubai, scheduled for completion in 2008, will be the world’s tallest building with a height of 705 m. The area of its ground footprint will be 8000 m2 . Convert its height and footprint area to U.S.customary units to three signiﬁcant digits.
h D 705 m A D 8000 m2 3.281 ft 1m 3.218 ft 1m D 2.31 ð 103 ft
D 8.61 ð 104 ft2
h D 2.31 ð 103 ft,
A D 8.61 ð 104 ft2
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Problem 1.6 Suppose that you have just purchased a Ferrari F355 coupe and you want to know whether you can use your set of SAE (U.S. Customary Units) wrenches to work on it. You have wrenches with widths w D 1/4 in, 1/2 in, 3/4 in, and 1 in, and the car has nuts with dimensions n D 5 mm, 10 mm, 15 mm, 20 mm, and 25 mm.Deﬁning a wrench to ﬁt if w is no more than 2% larger than n, which of your wrenches can you use?
Solution: Convert the metric size n to inches, and compute the
percentage difference between the metric sized nut and the SAE wrench. The results are: 5 mm 1 inch 25.4 mm D 0.19685.. in, 0.19685 0.25 0.19685 D 10 mm 1 inch 25.4 mm 1 inch 25.4 mm 1 inch 25.4 mm 1 inch 25.4 mm D 0.3937.. in, 27.0%100 D 27.0% 100
0.3937 0.5 0.3937 0.5905 0.5 0.5905 0.7874 0.75 0.7874 0.9843 1.0 0.9843
D 0.5905.. in,
100 D C15.3%
20 mm D 0.7874.. in, 100 D C4.7%
D 0.9843.. in,
A negative percentage implies that the metric nut is smaller than the SAE wrench; a positive percentage means that the nut is larger then the wrench. Thus within the deﬁnition ofthe 2% ﬁt, the 1 in wrench will ﬁt the 25 mm nut. The other wrenches cannot be used.
Problem 1.7 Suppose that the height of Mt. Everest is known to be between 29,032 ft and 29,034 ft. Based on this information, to how many signiﬁcant digits can you express the height (a) in feet? (b) in meters?.
a) h1 D 29032 ft h2 D 29034 ft The two heights are equal if rounded off to four...