CHAPTER 1: Introduction, Measurement, Estimating Responses to Questions 1. (a) A particular person’s foot. Merits: reproducible. Drawbacks: not accessible to the general public; not invariable (could change size with age, time of day, etc.); not indestructible. (b) Any person’s foot. Merits: accessible. Drawbacks: not reproducible (different people have different size feet); not invariable (could change size with age, time of day, etc.); not indestructible. Neither of these options would make a good standard. 2. The number of digits you present in your answer should represent the precision with which you know a measurement; it says very little about the accuracy of the measurement. For example, if you measure the length of a table to great precision, but with a measuring instrument that is not calibrated correctly, you will not measure accurately. 3. The writers of the sign converted 3000 ft to meters without taking significant figures into account. To be consistent, the elevation should be reported as 900 m. 4. The distance in miles is given to one significant figure and the distance in kilometers is given to five significant figures! The figure in kilometers indicates more precision than really exists or than is meaningful. The last digit represents a distance on the same order of magnitude as the car’s length! 5. If you are asked to measure a flower bed, and you report that it is “four,” you haven’t given enough information for your answer to be useful. There is a large difference between a flower bed that is 4 m long and one that is 4 ft long. Units are necessary to give meaning to the numerical answer. 6. Imagine the jar cut into slices each about the thickness of a marble. By looking through the bottom of the jar, you can roughly count how many marbles are in one slice. Then estimate the height of the jar in slices, or in marbles. By symmetry, we assume that all marbles are the same size and shape. Therefore the total number of marbles in the jar will be the product of the number of marbles per slice and the number of slices. 7. You should report a result of 8.32 cm. Your measurement had three significant figures. When you multiply by 2, you are really multiplying by the integer 2, which is exact. The number of significant figures is determined by your measurement. 8. The correct number of significant figures is three: sin 30.0º = 0.500. 9. You only need to measure the other ingredients to within 10% as well. 10. Useful assumptions include the population of the city, the fraction of people who own cars, the average number of visits to a mechanic that each car makes in a year, the average number of weeks a mechanic works in a year, and the average number of cars each mechanic can see in a week. (a) There are about 800,000 people in San Francisco. Assume that half of them have cars. If each of these 400,000 cars needs servicing twice a year, then there are 800,000 visits to mechanics in a year. If mechanics typically work 50 weeks a year, then about 16,000 cars would need to be seen each week. Assume that on average, a mechanic can work on 4 cars per day, or 20 cars a week. The final estimate, then, is 800 car mechanics in San Francisco. (b) Answers will vary. © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 1 Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual 11. One common way is to observe Venus at a Sun Venus time when a line drawn from Earth to Venus is perpendicular to a line connecting Venus to the Sun. Then Earth, Venus, and the Sun are at the vertices of a right triangle, with Venus at the 90º angle. (This configuration will result in the greatest angular distance between Venus and the Sun, as seen from Earth Earth.) One can then measure the distance to Venus, using radar, and measure the angular distance between Venus and the Sun. From this information you can use trigonometry to calculate the length of the leg of the triangle that is the distance from Earth to the Sun. 12. No. Length must be included as a base quantity. Solutions to Problems 1. (a) 14 billion years= 1.4×1010years (b) (1.4×1010y)(3.156×107s 1 y)= 4.4×1017s 2. (a) 214 3 significant figures (b) 81.60 4 significant figures (c) 7.03 3 significant figures (d) 0.03 1 significant figure (e) 0.0086 2 significant figures (f) 3236 4 significant figures (g) 8700 2 significant figures 3. (a) 1.156= 1.156×100 (b) 21.8= 2.18×101 (c) 0.0068= 6.8×10−3 (d) 328.65= 3.2865×102 (e) 0.219= 2.19×10−1 (f) 444= 4.44×102 4. (a) 8.69×104 = 86,900 (b) 9.1×103 = 9,100 (c) 8.8×10−1 = 0.88 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2 Chapter 1 Introduction, Measurement, Estimating (d) 4.76×102 = 476 (e) 3.62×10−5 = 0.0000362 0.25 m 5. % uncertainty = ×100% = 4.6% 5.48 m 0.2 s 6. (a) % uncertainty = ×100% = 4% 5 s 0.2 s (b) % uncertainty = ×100% = 0.4% 50 s 0.2 s (c) % uncertainty = ×100% = 0.07% 300 s 7. To add values with significant figures, adjust all values to be added so that their exponents are all the same. (9.2×103s)+(8.3×104s)+(0.008×106s)=(9.2×103s)+(83×103s)+(8×103s) =(9.2+83+8)×103s=100.2×103s= 1.00×105s When adding, keep the least accurate value, and so keep to the “ones” place in the last set of parentheses. 8. (2.079×102m)(0.082×10−1)= 1.7m. When multiplying, the result should have as many digits as the number with the least number of significant digits used in the calculation. 9. θ (radians) sin(θ ) tan(θ ) 0 0.00 0.00 Keeping 2 significant figures in the angle, and 0.10 0.10 0.10 expressing the angle in radians, the largest angle that has 0.12 0.12 0.12 the same sine and tangent is 0.24 radians. In degrees, 0.20 0.20 0.20 the largest angle (keeping 2 significant figure) is 12°. 0.24 0.24 0.24 The spreadsheet used for this problem can be found on 0.25 0.25 0.26 the Media Manager, with filename “PSE4_ISM_CH01.XLS,” on tab “Problem 1.9.” 10. To find the approximate uncertainty in the volume, calculate the volume for the minimum radius and the volume for the maximum radius. Subtract the extreme volumes. The uncertainty in the volume is then half this variation in volume. V = 4πr3 = 4π(0.84m)3 =2.483m3 specified 3 specified 3 V = 4πr3 = 4π(0.80m)3 =2.145m3 min 3 min 3 V = 4πr3 = 4π(0.88m)3 =2.855m3 max 3 max 3 ΔV = 1(V −V )= 1(2.855m3 −2.145m3)=0.355m3 2 max min 2 ΔV 0.355m3 The percent uncertainty is = ×100=14.3≈ 14%. V 2.483m3 specified © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 3 Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual 11. (a) 286.6 mm 286.6×10−3m 0.2866 m (b) 85μV 85×10−6V 0.000085 V (c) 760 mg 760×10−6kg 0.00076 kg (if last zero is not significant) (d) 60.0 ps 60.0×10−12s 0.0000000000600 s (e) 22.5 fm 22.5×10−15m 0.0000000000000225 m (f) 2.50 gigavolts 2.5×109volts 2,500,000,000 volts 12. (a) 1×106volts 1 megavolt =1 Mvolt (b) 2×10−6meters 2 micrometers =2μm (c) 6×103days 6 kilodays =6 kdays (d) 18×102bucks 18 hectobucks =18 hbucks or 1.8 kilobucks (e) 8×10−8seconds 80 nanoseconds =80 ns 13. Assuming a height of 5 feet 10 inches, then 5'10"=(70 in)(1 m 39.37 in)= 1.8 m. Assuming a weight of 165 lbs, then (165 lbs)(0.456 kg 1 lb) = 75.2 kg. Technically, pounds and mass measure two separate properties. To make this conversion, we have to assume that we are at a location where the acceleration due to gravity is 9.80 m/s2. 14. (a) 93 million miles =(93×106miles)(1610 m 1 mile)= 1.5×1011m (b) 1.5×1011m =150×109m = 150 gigameters or 1.5×1011m =0.15×1012m = 0.15 terameters 0.111 yd2 15. (a) 1 ft2 =(1 ft2)(1 yd 3 ft)2 =0.111 yd2, and so the conversion factor is . 1 ft2 10.8ft2 (b) 1 m2 =(1 m2)(3.28 ft 1 m)2 =10.8 ft2, and so the conversion factor is . 1m2 16. Use the speed of the airplane to convert the travel distance into a time. d = vt, so t = d v. ⎛ 1 h ⎞⎛3600s⎞ t = d v =1.00 km⎜ ⎟⎜ ⎟ = 3.8s ⎝950 km⎠⎝ 1 h ⎠ 17. (a) 1.0×10−10m =(1.0×10−10m)(39.37 in 1 m)= 3.9×10−9in ⎛ 1 m ⎞⎛ 1 atom ⎞ (b) (1.0 cm)⎜ ⎟⎜ ⎟= 1.0×108atoms ⎝100 cm⎠⎝1.0×10−10m⎠ © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4 Chapter 1 Introduction, Measurement, Estimating 18. To add values with significant figures, adjust all values to be added so that their units are all the same. 1.80 m+142.5 cm+5.34×105μm=1.80 m+1.425 m+0.534 m=3.759 m= 3.76 m When adding, the final result is to be no more accurate than the least accurate number used. In this case, that is the first measurement, which is accurate to the hundredths place when expressed in meters. ⎛0.621 mi⎞ 0.621mi h 19. (a) (1km h)⎜ ⎟=0.621mi h, and so the conversion factor is . ⎝ 1 km ⎠ 1km h ⎛3.28 ft⎞ 3.28ft s (b) (1m s)⎜ ⎟ =3.28ft s, and so the conversion factor is . ⎝ 1 m ⎠ 1m s ⎛1000 m⎞⎛ 1 h ⎞ 0.278m s (c) (1km h)⎜ ⎟⎜ ⎟=0.278m s, and so the conversion factor is . ⎝ 1 km ⎠⎝3600 s⎠ 1km h 20. One mile is 1.61×103m. It is 110 m longer than a 1500-m race. The percentage difference is calculated here. 110 m ×100% = 7.3% 1500 m 21. (a) Find the distance by multiplying the speed times the time. 1.00 ly = (2.998×108m s)(3.156×107s)= 9.462×1015m ≈ 9.46×1015m (b) Do a unit conversion from ly to AU. ⎛9.462×1015m⎞⎛ 1 AU ⎞ (1.00 ly)⎜ ⎟⎜ ⎟= 6.31×104AU ⎝ 1.00 ly ⎠⎝1.50×1011m⎠ ⎛ 1 AU ⎞⎛3600 s⎞ (c) (2.998×108m s)⎜ ⎟⎜ ⎟= 7.20AU h ⎝1.50×1011m⎠⎝ 1 hr ⎠ 1char 1min 1hour 1day 1year 22. (82×109bytes)× × × × × =2598 years≈ 2600years 1byte 180char 60min 8hour 365.25days 23. The surface area of a sphere is found by A= 4πr2 = 4π(d 2)2 =πd2. (a) A =πD2 =π(3.48×106m)2 = 3.80×1013m2 Moon Moon A πD2 ⎛ D ⎞2 ⎛ R ⎞2 ⎛6.38×106m⎞2 (b) Earth = Earth =⎜ Earth ⎟ =⎜ Earth ⎟ =⎜ ⎟ = 13.4 A πD2 ⎝D ⎠ ⎝R ⎠ ⎝1.74×106m⎠ Moon Moon Moon Moon 24. (a) 2800=2.8×103 ≈1×103 = 103 (b) 86.30×102 =8.630×103 ≈10×103 = 104 (c) 0.0076=7.6×10−3 ≈10×10−3 = 10−2 (d) 15.0×108 =1.5×109 ≈1×109 = 109 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 5 Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual 25. The textbook is approximately 25 cm deep and 5 cm wide. With books on both sides of a shelf, the shelf would need to be about 50 cm deep. If the aisle is 1.5 meter wide, then about 1/4 of the floor space is covered by shelving. The number of books on a single shelf level is then ⎛ 1 book ⎞ 1(3500m2)⎜ ⎟=7.0×104books. With 8 shelves of books, the total number of 4 ⎝(0.25 m)(0.05 m)⎠ books stored is as follows. ⎛ books ⎞ ⎜7.0×104 ⎟(8 shelves)≈ 6×105 books ⎝ shelf level⎠ 26. The distance across the United States is about 3000 miles. (3000 mi)(1 km 0.621 mi)(1 hr 10 km)≈ 500 hr Of course, it would take more time on the clock for the runner to run across the U.S. The runner could obviously not run for 500 hours non-stop. If they could run for 5 hours a day, then it would take about 100 days for them to cross the country. 27. A commonly accepted measure is that a person should drink eight 8-oz. glasses of water each day. That is about 2 quarts, or 2 liters of water per day. Approximate the lifetime as 70 years. (70 y)(365 d 1 y)(2 L 1 d)≈ 5×104L 28. An NCAA-regulation football field is 360 feet long (including the end zones) and 160 feet wide, which is about 110 meters by 50 meters, or 5500 m2. The mower has a cutting width of 0.5 meters. Thus the distance to be walked is as follows. area 5500m2 d = = =11000 m =11 km width 0.5 m At a speed of 1 km/hr, then it will take about 11 h to mow the field. 29. In estimating the number of dentists, the assumptions and estimates needed are: the population of the city the number of patients that a dentist sees in a day the number of days that a dentist works in a year the number of times that each person visits the dentist each year We estimate that a dentist can see 10 patients a day, that a dentist works 225 days a year, and that each person visits the dentist twice per year. (a) For San Francisco, the population as of 2001 was about 1.7 million, so we estimate the population at two million people. The number of dentists is found by the following calculation. ⎛ visits ⎞ ⎛ ⎞ 2 ⎜ ⎟ ⎜ ⎟ year ⎛ 1 yr ⎞ 1 dentist (2×106people)⎜ ⎟⎜ ⎟⎜ ⎟≈ 1800 dentists ⎜⎜1 person⎟⎟⎝225 workdays⎠⎜⎜10 visits ⎟⎟ ⎝ ⎠ ⎝ workday ⎠ (b) For Marion, Indiana, the population is about 50,000. The number of dentists is found by a similar calculation to that in part (a), and would be 45 dentists. There are about 50 dentists listed in the 2005 yellow pages. 30. Assume that the tires last for 5 years, and so there is a tread wearing of 0.2 cm/year. Assume the average tire has a radius of 40 cm, and a width of 10 cm. Thus the volume of rubber that is becoming pollution each year from one tire is the surface area of the tire, times the thickness per year © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6 Chapter 1 Introduction, Measurement, Estimating that is wearing. Also assume that there are 1.5×108 automobiles in the country – approximately one automobile for every two people. And there are 4 tires per automobile. The mass wear per year is given by the following calculation. ⎛mass⎞ ⎛surface area⎞⎛thickness wear⎞ ⎜ ⎟=⎜ ⎟⎜ ⎟(density of rubber)(# of tires) ⎝ year ⎠ ⎝ tire ⎠⎝ year ⎠ ⎡2π(0.4m)(0.1m)⎤ = (0.002m y)(1200kg m3)(6.0×108tires)= 4×108kg y ⎢ ⎥ ⎣ 1 tire ⎦ 31. Consider the diagram shown (not to scale). The balloon is a distance h above the d h surface of the Earth, and the tangent line from the balloon height to the surface of the earth indicates the location of the horizon, a distance d away from the balloon. r Use the Pythagorean theorem. r (r+h)2 =r2 +d2 → r2 +2rh+h2 =r2 +d2 2rh+h2 =d2 → d = 2rh+h2 d = 2(6.4×106m)(200m)+(200m)2 =5.1×104m≈ 5×104m (≈80 mi) 32. At $1,000 per day, you would earn $30,000 in the 30 days. With the other pay method, you would get $0.01(2t−1) on the tth day. On the first day, you get $0.01(21−1)=$0.01. On the second day, you get $0.01(22−1)=$0.02. On the third day, you get $0.01(23−1)=$0.04. On the 30th day, you get $0.01(230−1)=$5.4×106, which is over 5 million dollars. Get paid by the second method. 33. In the figure in the textbook, the distance d is perpendicular to the vertical radius. Thus there is a right triangle, with legs of d and R, and a hypotenuse of R+h. Since h(cid:19) R, h2 (cid:19)2Rh. d2 +R2 =(R+h)2 = R2 +2Rh+h2 → d2 =2Rh+h2 → d2 ≈2Rh → d2 (4400m)2 R= = = 6.5×106m 2h 2(1.5m) A better measurement gives R =6.38×106m. 34. To see the Sun “disappear,” your line of sight to the top of the Sun is tangent to the Earth’s surface. Initially, you To 1st sunset d h are lying down at point A, and you see the first sunset. θ A Then you stand up, elevating your eyes by the height h. B To 2nd sunset While standing, your line of sight is tangent to the Earth’s surface at point B, and so that is the direction to R the second sunset. The angle θ is the angle through R which the Sun appears to move relative to the Earth θ during the time to be measured. The distance d is the distance from your eyes when standing to point B. Use the Pythagorean theorem for the following Earth center relationship. d2 +R2 =(R+h)2 = R2 +2Rh+h2 → d2 =2Rh+h2 © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 7 Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual The distance h is much smaller than the distance R, and so h2 (cid:19)2Rh which leads to d2 ≈2Rh. We also have from the same triangle that d R = tanθ, and so d = Rtanθ. Combining these two 2h relationships gives d2 ≈2Rh = R2tan2θ, and so R = . tan2θ The angle θ can be found from the height change and the radius of the Earth. The elapsed time between the two sightings can then be found from the angle, knowing that a full revolution takes 24 hours. 2h 2h 2(1.3m) R = → θ= tan−1 = tan−1 =(3.66×10−2)o tan2θ R 6.38×106m θ tsec = → 360o 3600s 24h× 1h ⎛ θ ⎞⎛ 3600s⎞ ⎛(3.66×10−2)o ⎞⎛ 3600s⎞ t =⎜ ⎟⎜24h× ⎟=⎜ ⎟⎜24h× ⎟= 8.8s ⎝360o ⎠⎝ 1h ⎠ ⎜ 360o ⎟⎝ 1h ⎠ ⎝ ⎠ mass units ⎡M ⎤ 35. Density units= = ⎢ ⎥ volume units ⎣L3 ⎦ 36. (a) For the equation v= At3 −Bt, the units of At3 must be the same as the units of v. So the units of A must be the same as the units of v t3 , which would be L T4 . Also, the units of Bt must be the same as the units of v. So the units of B must be the same as the units of v t, which would be L T2 . (b) For A, the SI units would be m s4 , and for B, the SI units would be m s2 . 37. (a) The quantity vt2 has units of (m s)(s2)=mis, which do not match with the units of meters for x. The quantity 2at has units (m s2)(s) = m s, which also do not match with the units of meters for x. Thus this equation cannot be correct. (b) The quantity v t has units of (m s)(s) = m, and 1at2 has units of (m s2)(s2)=m. Thus, 0 2 since each term has units of meters, this equation can be correct. (c) The quantity v t has units of (m s)(s) = m, and 2at2 has units of (m s2)(s2)= m. Thus, 0 since each term has units of meters, this equation can be correct. ⎡ L3 ⎤⎡ML2⎤ ⎢ ⎥⎢ ⎥ Gh ⎣MT2⎦⎣ T ⎦ ⎡L3L2T5M ⎤ ⎡T5⎤ 38. t = → = ⎢ ⎥ = ⎢ ⎥ = ⎡⎣T2⎤⎦ =[T] P c5 ⎡L⎤5 ⎣ MT3L5 ⎦ ⎣T3⎦ ⎢ ⎥ ⎣T⎦ © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 8 Chapter 1 Introduction, Measurement, Estimating 2 m 39. The percentage accuracy is ×100% = 1×10−5%. The distance of 20,000,000 m needs to 2×107m be distinguishable from 20,000,002 m, which means that 8 significant figures are needed in the distance measurements. 40. Multiply the number of chips per wafer times the number of wafers that can be made from a cylinder. ⎛ chips ⎞⎛ 1 wafer ⎞⎛ 250 mm ⎞ chips ⎜100 ⎟⎜ ⎟⎜ ⎟= 83,000 ⎝ wafer⎠⎝0.300 mm⎠⎝1 cylinder⎠ cylinder ⎛3.156×107s⎞ 41. (a) # of seconds in 1.00 y: 1.00 y=(1.00 y)⎜ ⎟= 3.16×107s ⎝ 1 y ⎠ ⎛3.156×107s⎞⎛1×109ns⎞ (b) # of nanoseconds in 1.00 y: 1.00 y=(1.00 y)⎜ ⎟⎜ ⎟= 3.16×1016ns ⎝ 1 y ⎠⎝ 1 s ⎠ ⎛ 1 y ⎞ (c) # of years in 1.00 s: 1.00 s=(1.00 s)⎜ ⎟= 3.17×10−8y ⎝3.156×107s⎠ 42. Since the meter is longer than the yard, the soccer field is longer than the football field. 1.09yd L −L =100m× −100yd= 9yd soccer football 1m 1m L −L =100m−100yd× = 8m soccer football 1.09yd Since the soccer field is 109 yd compare to the 100-yd football field, the soccer field is 9% longer than the football field. 43. Assume that the alveoli are spherical, and that the volume of a typical human lung is about 2 liters, which is .002 m3. The diameter can be found from the volume of a sphere, 4πr3. 3 πd3 4πr3 = 4π(d 2)3 = 3 3 6 d3 ⎡6(2×10−3) ⎤1/3 (3×108)π =2×10−3m3 → d = ⎢ m3⎥ = 2×10−4m 6 ⎢⎣ 3×108π ⎥⎦ ⎛1.000×104m2 ⎞⎛3.281ft⎞2⎛ 1acre ⎞ 44. 1 hectare =(1 hectare)⎜ ⎟⎜ ⎟ ⎜ ⎟= 2.471acres ⎝ 1hectare ⎠⎝ 1m ⎠ ⎝4.356×104ft2 ⎠ 45. There are about 3×108 people in the United States. Assume that half of them have cars, that they each drive 12,000 miles per year, and their cars get 20 miles per gallon of gasoline. ⎛1 automobile⎞⎛12,000 mi auto⎞⎛1 gallon⎞ (3×108people)⎜ ⎟⎜ ⎟⎜ ⎟≈ 1×1011gal y ⎝ 2 people ⎠⎝ 1 y ⎠⎝ 20 mi ⎠ © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 9 Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual ⎛ 10−15kg ⎞⎛1 proton or neutron⎞ 46. (a) ⎜ ⎟⎜ ⎟= 1012protons or neutrons ⎝1 bacterium⎠⎝ 10−27kg ⎠ ⎛ 10−17kg ⎞⎛1 proton or neutron⎞ (b) ⎜ ⎟⎜ ⎟= 1010protons or neutrons ⎝1 DNA molecule⎠⎝ 10−27kg ⎠ ⎛ 102kg ⎞⎛1 proton or neutron⎞ (c) ⎜ ⎟⎜ ⎟= 1029protons or neutrons ⎝1 human⎠⎝ 10−27kg ⎠ ⎛ 1041kg ⎞⎛1 proton or neutron⎞ (d) ⎜ ⎟⎜ ⎟= 1068protons or neutrons ⎝1 galaxy⎠⎝ 10−27kg ⎠ 47. The volume of water used by the people can be calculated as follows: ⎛1200L day⎞⎛365day⎞⎛1000cm3⎞⎛ 1km ⎞3 (4×104people)⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ =4.38×10−3km3 y ⎝ 4 people ⎠⎝ 1 y ⎠⎝ 1L ⎠⎝105cm⎠ The depth of water is found by dividing the volume by the area. V 4.38×10−3km3 y ⎛ km⎞⎛105cm⎞ d = = =⎜8.76×10−5 ⎟⎜ ⎟=8.76cm y≈ 9cm y A 50 km2 ⎝ y ⎠⎝ 1 km ⎠ 48. Approximate the gumball machine as a rectangular box with a square cross-sectional area. In counting gumballs across the bottom, there are about 10 in a row. Thus we estimate that one layer contains about 100 gumballs. In counting vertically, we see that there are about 15 rows. Thus we estimate that there are 1500 gumballs in the machine. 49. Make the estimate that each person has 1.5 loads of laundry per week, and that there are 300 million people in the United States. 1.5 loads week 52weeks 0.1kg kg kg (300×106people)× × × =2.34×109 ≈ 2×109 1person 1 y 1load y y ⎛3V ⎞1/3 50. The volume of a sphere is given by V = 4πr3, and so the radius is r =⎜ ⎟ . For a 1-ton rock, 3 ⎝4π⎠ the volume is calculated from the density, and then the diameter from the volume. ⎛2000 lb⎞⎛ 1ft3 ⎞ V =(1 T)⎜ ⎟⎜ ⎟=10.8 ft3 ⎝ 1 T ⎠⎝186 lb⎠ ⎛3V ⎞1/3 ⎡3(10.8 ft3)⎤1/3 d =2r =2⎜ ⎟ =2⎢ ⎥ =2.74 ft ≈ 3 ft ⎝4π⎠ ⎢⎣ 4π ⎥⎦ 8bits 1sec 1min 51. (783.216×106bytes)× × × =74.592min ≈ 75min 1byte 1.4×106bits 60sec © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 10