Construccion

Problem 1.1 The value of is 3.14159265. . . If C is the circumference of a circle and r is its radius, determine the value of r/C to four significant digits.

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?

Solution:
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
2

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.

Solution:
h D 705 m A D 8000 m2 3.281 ft 1m 3.218 ft 1m D 2.31 ð 103 ft
2

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

15 mm

D 0.5905.. in,

100 D C15.3%

n
20 mm D 0.7874.. in, 100 D C4.7%

25 mm

D 0.9843.. in,

100 D

1.6%

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?.

Solution:
a) h1 D 29032 ft h2 D 29034 ft The two heights are equal if rounded off to four...