Solved: (Note: Remember that for rough estimates, only round numbers are needed both as

(I) The age of the universe is thought to be about 14 billion years. Assuming two significant figures, write this in powers of ten in (a) years. (6) seconds.

(I) How many significant figures do each of the following numbers have: (rr) 214, (b) 860. (c) 7.03. (d) 0.03. (e) 0.0086, (/) 3236. and {g) 8700?

(I) Write the following numbers in powers of ten notation: () 156, (b) 28, (c) 0.0068, (d) 27.635. (e) 0.219, and {f ) 444.

(I) Write out the following numbers in full with the correct number of zeros: (<?) 8.69 x 104, (b) 9.1 X 103, (c) 8.8 x 10"1, (d) 4.76 x 102, and (e) 3.62 x 10~5.

(II) What, approximately, is the percent uncertainty for the measurement given as 57 nr?

(II) What is the percent uncertainty in the measurement 3.76 0.25 m ?

(II) Time intervals measured with a stopwatch typically have an uncertainty of about 0.2 s, due to human reaction time at the start and stop moments. What is the percent uncertainty of a handtimed measurement of (a) 5 s, (/>) 50 s. (c) 5 min?

(II) Add (9.2 X lO^s) + (8.3 X 104s) + (0.008 X 10fts).

(II) Multiply 2.079 X 102m by 0.082 x 10 . taking into account significant figures.

(Ill) What is the area, and its approximate uncertainty, of a circle of radius 3.8 x 104cm?

(Ill) What, roughly, is the percent uncertainty in the volume of a spherical beach ball whose radius is r = 2.86 0.09 m?

(I) Write the following as full (decimal) numbers with standard units: (a) 286.6 mm, (b) 85juV. (c) 760 mg, {d) 60.0 ps. (e) 22.5 fm. (/) 2.50 gigavolts.

(I) Express the following using the prefixes of Table 1-4: (r/) 1x106 volts. (b) 2 x 106 meters, (c) 6x103days, (d) 18x102 bucks, and (e) 8 X 10-9 pieces.

(I) Determine your own height in meters, and your mass in kg.

(I) The Sun. on average, is 93 million miles from Earth. How many meters is this? Express (a) using powers of ten. and (b) using a metric prefix.

(II) What is the conversion factor between (tf) ft2 and vd2. (b) m2 and ft2?

(II) An airplane travels at 950 km/h. How long does it take to travel 00 km?

(II) A typical atom has a diameter of about 0 X 10_,"m. (a) What is this in inches? (b) Approximately how many atoms are there along a 0-cm line?

(II) Express the following sum with the correct number of significant figures: 80 m + 142.5 cm + 5.34 x 105jum.

(II) Determine the conversion factor between (a) km/h and mi/h. (b) m/s and ft/s. and (c) km/h and m/s.

(II) How much longer (percentage) is a one-mile race than a 1500-m race ("the metric mile)?

(II) A light-year is the distance light travels in one year (at speed = 2.998 X 108m/s). (a) How many meters are there in 00 light-year? (b) An astronomical unit (AU) is the average distance from the Sun to Earth, 50x10s km. How many AU are there in 00 light- year? (c) What is the speed of light in AU/h?

(Ill) The diameter of the Moon is 3480 km. (a) What is the surface area of the Moon? (b) How many times larger is the surface area of the Earth?

(I) Estimate the order of magnitude (power of ten) of: (<r) 2800. (b) 86.30 x 102. (c) 0.0076. and (d) 0 x 10s.

(II) Estimate how many books can be shelved in a college library with 3500 square meters of floor space. Assume 8 shelves high, having books on both sides, with corridors 5 m wide. Assume books are about the size of this one. on average.

(II) Estimate how many hours it would take a runner to run (at 10 km/h) across the United States from New York to California.

(II) Estimate how long it would take one person to mow a football field using an ordinary home lawn mower (Fig. 1-13). Assume the mower moves with a 1 km/h speed, and has a 0.5 m width.

(II) Estimate the number of liters of water a human drinks in a lifetime.

(II) Make a rough estimate of the volume of your body (in cm3).

(II) Make a rough estimate, for a typical suburban house, of the % of its outside wall area that consists of window- area.

(Ill) The rubber worn from tires mostly enters the atmosphere as particulate pollution. Estimate how much rubber (in kg) is put into the air in the United States every year. To get started, a good estimate for a tire treads depth is 1 cm when new. and the density of rubber is about 1200 kg/m3.

(II) The speed. of an object is given by the equation v = Ar Bts where / refers to time. What are the dimensions of A and B?

(II) Three students derive the following equations in which .v refers to distance traveled, v the speed, a the acceleration (m/s2). andtthe time, and the subscript (o) means a quantity at time t = 0:(a) x=vt2+ 2at. (b) x=VQt+ i*/2. and (c)x = l^f + lat2.Which of these could possibly be correct according to a dimensional check?

Global positioning satellites (GPS) can be used to deter- mine positions with great accuracy. The system works by determining the distance between the observer and each of several satellites orbiting Earth. If one of the satellites is at a distance of 20.000 km from you, what percent accu- racy in the distance is required if wre desire a 2-meter uncertainty? How- many significant figures do we need to have in the distance?

Computer chips (Fig. 1-14) are etched on circular silicon wafers of thickness 0.60 mm that are sliced from a solid cylindrical silicon crystal of length 30 cm. If each wafer can hold 100 chips. wrhat is the maximum number of chips that can be produced from one entire cylinder? FIGURE 1-14Problem 35. The wafer held by the hand (above) is shown below, enlarged and illuminated In- colored light. Visible arc rows of integrated circuits (chips).

(a) How many seconds are there in 00 year? (b) How many nanoseconds are there in 00 year? (c) How many years are there in 00 second?

A typical adult human lung contains about 300 million tiny cavities called alveoli. Estimate the average diameter of a single alveolus.

One hectare is defined as 104m2. One acre is 4 X 104 ft2. How many acres are in one hectare?

Use Table 1-3 to estimate the total number of protons or neutrons in (a) a bacterium. (b) a DNA molecule, (c) the human body, (d) our Galaxy.

Estimate the number of gallons of gasoline consumed by the total of all automobile drivers in the United States, per year.

Estimate the number of gumballs in the machine of Fig. 1-

An average family of four uses roughly 1200 liters (about 300 gallons) of water per day. (One liter = 1000 cm3.) How much depth would a lake lose per year if it uniformly covered an area of 50 square kilometers and supplied a local town with a population of 40.000 people? Consider only population uses, and neglect evaporation and so on.

How big is a ton? That is. what is the volume of something that weighs a ton? To be specific, estimate the diameter of a 1-ton rock, but first make a wild guess: will it be 1 ft across, 3 ft. or the size of a car? [Hint: Rock has mass per volume about 3 times that of water, which is 1 kg per liter (103 cm3) or 62 lb per cubic foot.)

A heavy rainstorm dumps 0 cm of rain on a city 5 km wide and 8 km long in a 2-h period. How many metric tons (l metric ton = 103kg) of water fell on the city? (1 cm3 of water has a mass of 1 gram = 10- 3kg.) How many gallons of water was this?

end just blocks out the Moon (Fig. 116). Make appropriate measurements to estimate the diameter of the Moon, given that the Earth-Moon distance is 3.8 X 105 km.

Estimate how many days it would take to walk around the world, assuming 10 h walking per day at 4 km/h.

Noahs ark was ordered to be 300 cubits long, 50 cubits wide, and 30 cubits high. Ibe cubit was a unit of measure equal to the length of a human forearm, elbow to the tip of the longest finger. Express the dimensions of Noahs ark in meters, and estimate its volume (m3).

One liter (1000 cm3) of oil is spilled onto a smooth lake. If the oil spreads out uniformly until it makes an oil slick just one molecule thick, with adjacent molecules just touching, estimate the diameter of the oil slick. Assume the oil molecules have a diameter of 2 X 10 10m.

Jean camps beside a wide river and wonders how wide it is. She spots a large rock on the bank directly across from her. She then walks upstream until she judges that the angle between her and the rock, which she can still see clearly, is now at an angle of 30 downstream (Fig. 1-17). Jean measures her stride to be about one yard long. The distance back to her camp is 120 strides. About how far across, both in yards and in meters, is the river?

A watch manufacturer claims that its watches gain or lose no more than 8 seconds in a year. How accurate is this watch, expressed as a percentage?

The diameter of the Moon is 3480 km. What is the volume of the Moon? How many Moons would be needed to create a volume equal to that of Earth?

An angstrom (symbol A) is a unit of length, defined as 10-10 m. which is on the order of the diameter of an atom. (a) How many nanometers are in 0 angstrom? (b) How many femtometers or fermis (the common unit of length in nuclear physics) are in 0 angstrom? (c) How many angstroms are in 0 meter? (r/) How many angstroms arc in 0 light-year (see Problem 22)?

Determine the percent uncertainty in 8. and in sin 0, when (n) 8 = 0 0.5. (b) 8 = 75.6 0.5.

If you began walking along one of Earths lines of longitude and walked until you had changed latitude by 1 minute of arc (there are 60 minutes per degree), how far would you have walked (in miles)? This distance is called a "nautical mile.

Problem 1P (Note: In Problems, assume a number like 6.4 is accurate to ?0.1; and 950 is ? 10 unless 950 is said to be ?precisely? or ?very nearly? 950, in which case assume 950 ? 1.) (II) The age of the universe is thought to be about 14 billion years. Assuming two significant figures, write this in powers of 10 in (a) years, (b) seconds.

Problem 1Q What are the merits and drawbacks of using a person’s foot as a standard? Consider both (a) a particular person’s foot, and (b) any person’s foot. Keep in mind that it is advantageous that fundamental standards be accessible (easy to compare to), invariable (do not change), indestructible, and reproducible.

Problem 2P (Note: In Problems, assume a number like 6.4 is accurate to ?0.1; and 950 is ? 10 unless 950 is said to be ?precisely? or ?very nearly? 950, in which case assume 950 ? 1.) (I) How many significant figures do each of the following numbers have: (a) 214, (b) 81.60, (c) 7.03, (d) 0.03, (e) 0.0086, (f) 3236, and (g) 8700?

Problem 2Q When traveling a highway in the mountains, you may see elevation signs that read “914 m (3000 ft).” Critics of the metric system claim that such numbers show the metric system is more complicated. How would you alter such signs to be more consistent with a switch to the metric system?

Problem 3P (Note: In Problems, assume a number like 6.4 is accurate to ± 0.1; and 950 is ±10 unless 950 is said to be “precisely” or “very nearly” 950, in which case assume 950 ± 1.) Write the following numbers in powers of ten notation: (a) 1.156, (b) 21.8, (c) 0.0068, (d) 27.635, (e) 0.219, and (f) 444

A heavy rainstorm dumps 1.0 cm of rain on a city 5 km wide and 8 km long in a 2-h period. How many metric tons (1 metric ton = 103 kg) of water fell on the city? [\(1\ \text{cm}^3\) of water has a mass of \(1\ \text{gram}=10^{-3}\ \text{kg}\).] How many gallons of water was this?

Hold a pencil in front of your eye at a position where its blunt end just blocks out the Moon (Fig. 1–16). Make appropriate measurements to estimate the diameter of the Moon, given that the Earth–Moon distance is 3.8 X 105 km.

Problem 46GP Estimate how many days it would take to walk around the world, assuming 10 h walking per day at 4 km/h.

Problem 47GP Noah’s ark was ordered to be 300 cubits long, 50 cubits wide, and 30 cubits high. The cubit was a unit of measure equal to the length of a human forearm, elbow to the tip of the longest finger. Express the dimensions of Noah’s ark in meters, and estimate its volume (m3).

Problem 48GP One liter (1000 cm3 )of oil is spilled onto a smooth lake. If the oil spreads out uniformly until it makes an oil slick just one molecule thick, with adjacent molecules just touching, estimate the diameter of the oil slick. Assume the oil molecules have a diameter of 2X 10-10 m.

Problem 3Q Why is it incorrect to think that the more digits you include in your answer, the more accurate it is?

Problem 4P (Note: In Problems, assume a number like 6.4 is accurate to ?0.1; and 950 is ? 10 unless 950 is said to be ?precisely? or ?very nearly? 950, in which case assume 950 ? 1.) (I) Write out the following numbers in full with the correct number of zeros: (a) 8.69 X 104 (b) 9.1 X 103, (c) 8.8 X 10-1 (d) 4.76 X 102 and (e) 3.62 X 10-5.

Jean camps beside a wide river and wonders how wide it is. She spots a large rock on the bank directly across from her. She then walks upstream until she judges that the angle between her and the rock, which she can still see clearly, is now at an angle of \(30^{\circ}\) downstream (Fig. 1-17). Jean measures her stride to be about one yard long. The distance back to her camp is 120 strides. About how far across, both in yards and in meters is the river? Figure 1-17 Equation Transcription: Text Transcription: 30 degrees

Problem 50GP A watch manufacturer claims that its watches gain or lose no more than 8 seconds in a year. How accurate are these watches, expressed as a percentage?

Problem 38GP One hectare is defined as 104 m2. One acre is 4 × 104 ft2. How many acres are in one hectare?

Problem 39GP Use Table 1–3 to estimate the total number of protons or neutrons in (a) a bacterium, (b) a DNA molecule, (c) the human body, (d) our Galaxy. Table 1–3 Some Masses Object Kilograms (approximate) Electron 10–30 kg Proton, neutron 10–2 kg DNA molecule 10–17 kg Bacterium 10–15 kg Mosquito 10–5 kg Plum 10–1 kg Human 102 kg Ship 108 kg Earth 6 × 1024 kg Sun 2 × 1030 kg Galaxy 1041 kg

Estimate the number of gumballs in the machine of Fig. 1-15. Figure 1-15

Problem 40GP Estimate the number of gallons of gasoline consumed by the total of all automobile drivers in the United States, per year.

Problem 42GP An average family of four uses roughly 1200 liters (about 300 gallons) of water per day. (One liter = 1000 cm3.) How much depth would a lake lose per year if it uniformly covered an area of 50 square kilometers and supplied a local town with a population of 40,000 people? Consider only population uses, and neglect evaporation and so on.

Problem 43GP How big is a ton? That is, what is the volume of something that weighs a ton? To be specific, estimate the diameter of a 1-ton rock, but first make a wild guess: will it be 1 ft across, 3 ft, or the size of a car? [Hint: Rock has mass per volume about 3 times that of water, which is 1 kg per liter (103 cm3) or 62 lb per cubic foot.]

Problem 4Q What is wrong with this road sign: Memphis 7 mi (11.263 km)?

Problem 5P (Note: Assume a number like 6.4 is accurate to 0.1; and 950 is 10 unless 950 is said to be “precisely” or “very nearly” 950, in which case assume 950 1.) (II) What, approximately, is the percent uncertainty for a measurement given as 1.57 m2?

Problem 5Q For an answer to be complete, the units need to be specified. Why?

Problem 6P (Note: In Problems, assume a number like 6.4 is accurate to ± 0.1; and 950 is ±10 unless 950 is said to be “precisely” or “very nearly” 950, in which case assume 950 ± 1.) What is the percent uncertainty in the measurement 3.76 ± 0.25 m?

Problem 6Q Discuss how the notion of symmetry could be used to estimate the number of marbles in a 1-liter jar.

Problem 51GP The diameter of the Moon is 3480 km. What is the volume of the Moon? How many Moons would be needed to create a volume equal to that of Earth?

Problem 52GP An angstrom (symbol Å) is a unit of length, defined as 10-10 which is on the order of the diameter of an atom. (a) How many nanometers are in 1.0 angstrom? (b) How many femtometers or fermis (the common unit of length in nuclear physics) are in 1.0 angstrom? (c) How many angstroms are in 1.0 m? (d) How many angstroms are in 1.0 light-year (see Problem 19)?

Determine the percent uncertainty in \(\theta\), and in \(\sin \ \theta\), when (a) \(\theta=15.0^{\circ} \pm 0.5^{\circ}\), (b) \(\theta=75.0^{\circ} \pm 0.5^{\circ}\) Equation Transcription: Text Transcription: theta sin theta theta = 15.0 degree pm 0.5 degree theta = 75.0 degree pm 0.5 degree

Problem 54GP If you began walking along one of Earth’s lines of longitude and walked until you had changed latitude by 1 minute of arc (there are 60 minutes per degree), how far would you have walked (in miles)? This distance is called a “nautical mile.”

Problem 7P (Note: In Problems, assume a number like 6.4 is accurate to ± 0.1; and 950 is ±10 unless 950 is said to be “precisely” or “very nearly” 950, in which case assume 950 ± 1.) Time intervals measured with a stopwatch typically have an uncertainty of about 0.2 s, due to human reaction time at the start and stop moments. What is the percent uncertainty of a handtimed measurement of (a) 5 s, (b) 50 s, (c) 5 min?

Problem 7Q You measure the radius of a wheel to be 4.16 cm. If you multiply by 2 to get the diameter, should you write the result as 8 cm or as 8.32 cm? Justify your answer.

Problem 8P (Note: In Problems, assume a number like 6.4 is accurate to ± 0.1; and 950 is ±10 unless 950 is said to be “precisely” or “very nearly” 950, in which case assume 950 ± 1.) Add (9.2 × 103 s) + (8.3 × 104 s) + (0.008 × 106 s).

Problem 8Q Express the sine of 30.0° with the correct number of significant figures.

Problem 9P (Note: In Problems, assume a number like 6.4 is accurate to ± 0.1; and 950 is ±10 unless 950 is said to be “precisely” or “very nearly” 950, in which case assume 950 ± 1.) Multiply 2.079 × 102 m by 0.082 × 10–1, taking into account significant figures.

Problem 9Q A recipe for a soufflé specifies that the measured ingredients must be exact, or the soufflé will not rise. The recipe calls for 6 large eggs. The size of “large” eggs can vary by 10%, according to the USDA specifications. What does this tell you about how exactly you need to measure the other ingredients?

Problem 10P (Note: In Problems, assume a number like 6.4 is accurate to ± 0.1; and 950 is ±10 unless 950 is said to be “precisely” or “very nearly” 950, in which case assume 950 ± 1.) What is the area, and its approximate uncertainty, of a circle of radius 3.8 × 104 cm?

Problem 11P (Note: In Problems, assume a number like 6.4 is accurate to ± 0.1; and 950 is ±10 unless 950 is said to be “precisely” or “very nearly” 950, in which case assume 950 ± 1.) What, roughly, is the percent uncertainty in the volume of a spherical beach ball whose radius is r= 2.86 ± 0.09 m?

Problem 12P Write the following as full (decimal) numbers with standard units: (a) 286.6 mm, (b) 85 ?V, (c) 760 mg, (d) 60.0 ps, (e) 22.5 fm, (f) 2.50 gigavolts.

Problem 10Q List assumptions useful to estimate the number of car mechanics in (a) San Francisco, (b) your hometown, and then make the estimates.

Express the following using the prefixes of Table 1–4: (a) \(1 \times 10^{6}\) volts (b) \(2\times10^{-6}\) meters (c) \(6 \times 10^{3}\) days (d) \(18 \times 10^{2}\) bucks and; (e) \(8 \times 10^{-9}\) pieces.

Problem 14P Determine your own height in meters, and your mass in kg.

Problem 15P (II) The Sun, on average, is 93 million miles from Earth. How many meters is this? Express (a) using powers of 10, and (b) using a metric prefix (km).

Problem 16P What is the conversion factor between (a) ft2 and yd2, (b) m2 and ft2?

Problem 17P An airplane travels at 950 km/h. How long does it take travel 1.00 km?

Problem 18P (II) A typical atom has a diameter of about 1.0 X 10-10 (a) What is this in inches? (b) Approximately how many atoms are along a 1.0-cm line, assuming they just touch?

Problem 20P (II) Determine the conversion factor between (a) km/h and mi/h (b) m/s and ft/s and (c) km/h and m/s.

(II) Express the following sum with the correct number of significant figures: 1.80 m + 142.5 cm + 5.34 x 105

Problem 21P (II) How much longer (percentage) is a one-mile race than a 1500-m race (“the metric mile”)?

Problem 22P (II) A light-year is the distance light travels in one year (at speed = 2.998 X 108 m/s)(a) How many meters are there in 1.00 light-year? (b) An astronomical unit (AU) is the average distance from the Sun to Earth, 1.50 X 108 km How many AU are there in 1.00 light-year?

Problem 23P The diameter of the Moon is 3480 km. (a) What is the surface area of the Moon? (b) How many times larger is the surface area of the Earth?

Problem 24P (Note: Remember that for rough estimates, only round numbers are needed both as input to calculations and as final results.) Estimate the order of magnitude (power of ten) of: (a) 2800, (b) 86.30 × 102, (c) 0.0076, and (d) 15.0 × 108.

Problem 25P (Note: Remember that for rough estimates, only round numbers are needed both as input to calculations and as final results.) (II) Estimate how many books can be shelved in a college library with3500 m2 of floor space. Assume 8 shelves high, having books on both sides, with corridors 1.5 m wide. Assume books are about the size of this one, on average.

Problem 26P (Note: Remember that for rough estimates, only round numbers are needed both as input to calculations and as final results.) (II) Estimate how many hours it would take to run (at 10 km/h ) across the U.S. from New York to California.

(II) Estimate how long it would take one person to mow a football field using an ordinary home lawn mower (Fig. 1-13). Assume the mower moves with a 1km/h speed, and has a 0.5 m width.

Problem 28P (Note: Remember that for rough estimates, only round numbers are needed both as input to calculations and as final results.) (II) Estimate the number of liters of water a human drinks in a lifetime.

Problem 29P (Note: Remember that for rough estimates, only round numbers are needed both as input to calculations and as final results.) Make a rough estimate of the volume of your body (in cm3).

Problem 30P (Note: Remember that for rough estimates, only round numbers are needed both as input to calculations and as final results.) Make a rough estimate, for a typical suburban house, of the % of its outside wall area that consists of window area.

Problem 31P (Note: Remember that for rough estimates, only round numbers are needed both as input to calculations and as final results.) The rubber worn from tires mostly enters the atmosphere as particulate pollution. Estimate how much rubber (in kg) is put into the air in the United States every year. To get started, a good estimate for a tire tread’s depth is 1 cm when new, and the density of rubber is about 1200 kg/m3.

Problem 32P The speed, v, of an object is given by the equation v = At3 – Bt, where t refers to time. What are the dimensions of A and B?

Problem 33P (II) Three students derive the following equations in which x refers to distance traveled, v the speed, a the acceleration (m/s2) t the time, and the subscript zero (0) means a quantity at time t =0 . Here are their equations: (a) x =vt2 +2at (b) x =v0 t +1/2 at2 and (c) x =v0t +2at2. Which of these could possibly be correct according to a dimensional check, and why?

Problem 34GP Global positioning satellites (GPS) can be used to determine positions with great accuracy. The system works by determining the distance between the observer and each of several satellites orbiting Earth. If one of the satellites is at a distance of 20,000 km from you, what percent accuracy in the distance is required if we desire a 2-meter uncertainty? How many significant figures do we need to have in the distance?

Computer chips (Fig. are etched on circular silicon wafers of thickness that are sliced from a solid cylindrical silicon crystal of length . If each wafer can hold 100 chips, what is the maximum number of chips that can be produced from one entire cylinder? FIGURE Problem 35 The wafer held by the hand (above) is shown below, enlarged and illuminated by colored light. Visible are rows of integrated circuits (chips).

Problem 36GP (a) How many seconds are there in 1.00 year? (b) How many nanoseconds are there in 1.00 year? (c) How many years are there in 1.00 second?

Problem 37GP A typical adult human lung contains about 300 million tiny cavities called alveoli. Estimate the average diameter of a single alveolus.