C. Selected Answers

Numerical answers are given as ranges or other hints, meant to facilitate checking for gross departures from the right track, without revealing the precise answer so that shortcuts are discouraged. Questions for which the answer is already known (questions asking to verify an answer), easily validated in the text, or that are a matter of original thought or opinion may not be included here.

The ranges sometimes may be annoyingly large, but think of them as guard rails to prevent a tragic miscalculation or to catch a fundamental misunderstanding of the underlying concepts. It can help catch errors like dividing the wrong things or swapping numerator and denominator, or multiplying when division is called for. In many cases, intuition, or guessing, might lead you already to similar answers or ranges. With practice, students may be able to anticipate what they think are reasonable ranges for answers. In fact, it is a great practice to think about expectations before working on the problem.

This appendix, then, might be thought of as an “intuition implant” that simulates how problems are for experts. Real life does not provide “answers at the back of the book,” so experts rely on experience, intuition, and a sense for “reasonable” results to help them understand when they’ve taken a wrong turn. A successful use of this appendix would help train students to develop their own “common sense” guard rails.

Chapter 1** 20. A smidge higher then boiling¶

21. Between 200 and 250 K

22. Between 250 and 300 years 22. Between 150 and 250 K

23. Between 250 and 300 years 23. Between 100 and 275 K

24. Between 10¹⁸ and 10²⁰

25. Between 100 and 150

Chapter 2¶

8. More than 10⁴⁰

9. Between 5 and 500 1. Nearly $100 billion

10. Later than 23:50 2. 4%: $\sim$$1 trillion; 5%: more than $100 trillion

11. Before 12:10 AM 4. On the low end of advanced countries

12. Between 10 and 40 years 5. Between 25 and 100 MJ/$

13. Between 75 and 100 years 6. Between 5% and 50%

14. Between 300 and 500 billion 7. The text had trading art, singing lessons,

15. It’s not 100 times longer therapy, and financial planning

16. A few millennia 8. Between 100 and 1,000; Less than 10 to go

17. Between 50 and 100 W 9. Between 20 and 100 years

10. May help to think of something once preva- **Chapter 4¶

lent, now rare

1. Earth: smaller than peppercorn and

2. Especially fruitful might be biological debasketball-court distant; Moon: sand grain

pendencies a hand’s width away

14. It can’t all be free of material substance 3. Comparable to the actual Earth radius

15. 1 AU = 1 km; Earth 1/12,000 km

Chapter 3¶

6. A fast walk or slow jog

7. Think about the subtended angle

8. Less than a third

9. Multiply sets to get accumulated scale fac-

10. Comparable to U.S. population today tors

11. Comparable to world population 200 years 9. A good deal farther than the moon, but still

ago well short of the sun/Mars

5. Table 3.2 offers a rough check 10. Ratio is more than a billion, and would

6. Over 16 billion; less than half the time we take more than 4 lifetimes

now experience 16. Think in terms of area as fraction of plot

7. Pretty close to Table 3.2 except for first two space

entries 17. Text has climbing Mt. Everest, supersonic

8. Answer must be less than 14 billion; whereas commercial flight, squirrel obstacle course,

Problem 6 was in excess of 15 billion and economic decoupling

9. Two are negative; three are positive 18. Will take 15–20 tanks of gas, and achieve a fuel economy a factor of 30 or so below typical

10. Between 1 and 5% cars

11. Add almost a half million; more than half 19. Double the gasoline from previous prob-

million born; less than half million died lem; gasoline mass almost as much as the car

14. Answers should round to the table valitself

ues

16. Only one country in the table creates more total demand, and only two have higher per-citizen contributions

17. Correct results are in the table

18. See Figure 3.15

19. This is why Africa gets attention, while North America is perhaps a greater concern.

20. It nearly triples

21. Area is key

22. Lesotho is relevant

Chapter 5** 29. Between 1 and 2 hours per day¶

30. The largest number is near 10⁸

31. Several inches 31. Six of the entries are inverses of six others

32. About the length of a typical room 32. A little over an Amp

33. A few kJ total, most in sliding 33. Nearly 10 kW; will cost over $1,000; don’t

34. You’ve got this do this!

35. Nearly 1 GJ 34. Will last 2–3 hours

36. Two of the points in Example 5.2.1 offer 35. In line with most chemical reactions, in

guidance the 50–200 kJ/mol range

8. Figure 5.1 offers hints 36. Between 3–$5\times 10^{-19}$ J per photon; get more than $10^{18}/$sec

9. Roughly half human metabolic power 37. In the neighborhood of 1 $\mu \mathrm{m}$ or 1 eV

10. Less than 5 seconds

11. Between 50–100 kcal (200–400 kJ)

13. Sensibly, a little less than 2 minutes **Chapter 6¶

14. Results should be roughly consistent with 1. Approx. 200 kJ, depending on mass

Figure 5.2 2. Several minutes

15. A bit less than 10% of household electricity 3. Several minutes

16. They’re actually close, within 10% 4. About 5 minutes

17. More than two 5. A few hours

18. Several kWh; less than $1 6. Not below freezing

19. On the low end of the human metabolism 7. Not quite up to “room” temperature

range; the equivalent cost of 10–20 burritos 8. Not quite half the time

20. A few hundred W 9. The cost of two burritos per day

21. Comparable to running a clothes dryer 10. Instances of heat/flame causing move-

(Fig. 5.2) ment

22. Several hundred MJ 11. See Table 6.2

23. Over 100 kWh; 2–4 burritos-worth 12. Roughly 30 kJ and 100 J/K

24. Several Therms; cost of fast-food lunch 13. Between 5–10%

25. Several gallons; cost of fast-food lunch for 14. A couple dozen percent, roughly

two 15. Pushing 100%, but not quite there

26. One is about twice the other 16. Achieves about 1/3 of theoretical

27. A little in excess of 10 kW 17. $\Delta T > 50^{\circ}\mathrm{C}$; environment not that cold

28. Less than a quarter of estimated 18. Close to a dozen kJ

29. Roughly one-third

10. 2 H per C plus 2 more

11. In the neighborhood of 20 bbl/yr

12. Should be appropriate fraction of 10,000 W total

13. a) between 30–40%; b) almost all; c) close to 13. A little over 100 MJ and a few dozen kWh

14. Sum to about 15 kg, which would fill a

15. Coal is near 12 qBtu, for instance refrigerator shelf in the water-bottle equivalent.

16. Nuclear is about 22%, for instance 15. drinking glass

17. Residential is about 5 qBtu, for instance 16. A few dozen times more volume, and about

18. Industry is a little over 30%, for instance 10² in mass

19. About 14% is renewable, for instance 17. $> 1, 000\times$ more expensive

18. Will cost nearly $1,000

19. Between 10–15%

23. Approximately half-century

24. Roughly a third

25. Surprisingly soon: maybe before student 25. If the rate of production increases…

26. What have you wanted that was all gone?

27. Shorter than R/P suggests

28. Pay attention to the dashed line 28. Opposite of virtual

29. Pay attention to the dashed line 29. Can’t have what’s not there

30. Reasons could fill a book

1. All lines overlap the up-slope

5. What enabled, then disappeared?

Chapter 9** 29. Sum to about 700 years; almost all in ice¶

and ocean

1. A single integer works okay for all three 30. A few hundred meters

2. Nearly 100 kg 31. A finger’s breadth per year

3. Between 10–15 kg 32. Keen to hear your thoughts

4. Approaching 1 GJ, and human-mass scale 33. Keen to hear your thoughts

5. Total is like small adult or large child

6. More than a factor of two

Chapter 10¶

7. Get about 50 years; rate not constant 1. Mostly clean; not all, though

8. The numbers basically match 2. Nothing is free

9. Between 1–2 ppm$_{\mathrm{v}}$, in agreement with Figure 9.3 3. What would unlimited mean?

10. What is it we know? 4. Table 10.2 has some help

11. Seems deserving of high marks 5. Can’t rely on any sun-driven energy

12. Historical vs. current activity levels 6. Between 200–250 $\mathrm{W/m}^{2}$

13. About $10^{\circ}\mathrm{C}$ cooler than actual 7. Photosynthesis supports essentially all life

14. Two pure cases and one partial 8. Comparing numbers in TW

15. Several degrees warmer 9. More than half

16. Very good for us at the right level 10. A little less than 1%

17. Numbers are not far from realistic 11. Not far from 1,000 $\mathrm{W/m}^{2}$

18. Triple pre-industrial and almost $5^{\circ}\mathrm{C}$ 12. Nearly 10 degrees

19. End $\sim 3^{\circ}\mathrm{C}$ high; almost linear, but not quite 13. Look for crazy-big input

20. No need to balance: Nature doesn’t 14. Between 0.5–1 gallon

bother 15. More than $4,000\times$

21. It’s no game-changer

22. Student’s choice

23. E.g., 390 $- 152 = 238$ for a match

24. Use 290.6 K; looks like continuation of panel progression

25. A few millimeters

26. A little over a century

27. A year or two

28. A couple of degrees

10. Runs approximately 10 kW to 1 MW

11. Roughly two-thirds the original speed

12. Close to 10 MW

13. About 10 stories of a building 13. Closer to 10 m/s than to 15 m/s

14. Almost double freeway speeds

15. About 4 times higher than airliners travel 15. Between 5–10 m/s

16. About two-thirds Earth radius 16. In the ballpark of 70 kW

17. Try using half the mass and half the en- 17. Recover 0.65%

18. Unpack $\mathrm{W/m}^{2}$ to confirm $\mathrm{kg/s}^{3}$

19. Cube is roughly as big as height from 19. Outer box area corresponds to running at 100%, full time

8. About 6 times typical nuclear plant 20. Definitely less than 50%

21. Looks like a factor of 8

22. Approaching (American) football field length

23. Approximately 1 MW

24. They may not have equivalent energy

25. Approaching 10,000 cubic meters per sec-needs

15. You’ve got a little over an hour

4. Mass shows up in both $mgh$ and $^{1}$

5. In the neighborhood of 1,500 m/s

6. About 5–10 humans–worth of mass!

7. Comparable to the height of Mt. Everest

Chapter 13** 26. Best at latitude; almost 15% better than¶

flat

1. How big are the packages? 27. Approaches 6 $\mathrm{kWh/m}^{2}/$day

2. Something times 10²¹ 28. Large house (and just the PV for one person)

3. Roughly 10¹⁶

4. Square is about as wide as Arizona or

5. About 1,000 California east-to-west

6. About 4,000 times 30. Cost, surely—but other challenges and

7. Use Eq. 13.3 to guide your reasoning mismatches as well

8. Should match Figure 13.1 31. A little over 200 W

9. One micron for each finger? 32. Roughly the size of a bedroom

10. Think about spill-over into UV and/or IR 33. Will spend a little over $4,000

11. Peak around $2.5 \times 10^{8}$, about 1 $\mu \mathrm{m}$ wide; 34. A little over a decade

matches up well 35. Even lower than $\sim 20\%$ from insolation vs.

11. Think energetics and depth overhead

12. Is the answer transparent? 36. About $2-worth of sun

13. Just comparing two energies 37. Hint: study Figures 13.23 and 13.24

14. Several hundred km/s 38. In absolute terms…

15. Condense the saga to that of a winner

16. Answer might involve physics, biology,

Chapter 14¶

rooftops

17. Inversely: larger in one means smaller in 1. About a dozen tons of CO$_{2}$

the other 2. Between 0.1–0.5%

18. Already extremely similar 3. Almost 100 logs per person per year

19. Think of current as a rate of electron flow 4. In the neighborhood of half-dozen logs per

in the circuit day

20. Get very close to 1,360 $\mathrm{W/m}^{2}$ 5. Won’t be exactly 15 years, but close

21. Sweltering is not preferred 6. Almost 1.5 L of ethanol

22. Between 5–6 $\mathrm{kWh/m}^{2}/$day; between 200– 7. Roughly consistent with Table 14.1 for coal

250 $\mathrm{W/m}^{2}$ 8. Net is one-third production

23. Involves interpreting $\mathrm{kWh/m}^{2}/$day as full-sun-hours 9. Extra land is twice the yield-land

24. Not far from 200 $\mathrm{W/m}^{2}$ 10. A bit longer than a U.S. Presidential term

25. Range straddles 200 $\mathrm{W/m}^{2}$, varying about 11. Nothing to spare

10% 12. Corn now approximately 15% as much as this; still more than total arable land

13. Box barely fits north–south in U.S. 28. From steam onwards, it’s basically the same

14. Personal preferences play a role 29. Between 3 and 5 cents per kWh

15. A few per week!

31. Around 20 tons per year (more in reality)

32. Almost 2 million tons

33. A few hundred tons

34. Less than a decade

37. Two stand out

38. Two diagonals have no gray squares 38. Centuries

39. One has a half-life longer than a million 39. More often than once every two years

41. A nearly exact match!

42. Roughly twice as old as agriculture 42. Worked out in text: no calculation necessary—just interpretation

9. One is about 3% of the other (both decay) 43. Energy jump size

44. Like a milk jug

45. That last step might take a while

Chapter 16¶

1. Shortfall is more than a factor of 200

2. Somewhere between a car and a bus? 2. A bit farther than the record

3. About as thick as a six-story building is

4. Around a couple-dozen micrograms tall

6. Ranges about 55–85%

7. Energy has a mass, via $E = mc^{2}$ 7. Geothermal is a bit less than 1% of alterna-

8. Adds about 1% to the mass tive electricity

9. A little shy of the 8 m design height, sensibly

9. Works out

10. Diameter like a small house’s footprint

11. Comparable to human metabolism; 1% of

12. $A$ is twice a prime number American demand

13. Stick to 80 $< A < 110$ and 125 $< A < 155$ 12. A little more than 6 times that in Example 16.4.1

Chapter 17** 8. Wait; who has my…¶

9. Focus on what has mattered until now

10. Algae who? 10. What’s the alternative?

11. Two words almost say it all 13. Please figure out how it can work!

12. Fine if it is a little shy: transfer rates vary 14. Is this the movie version, or the real-life

13. Think about what a house can access, and one?

steam plants

10. May be up there with solar (4 to 6, likely)

Chapter 20¶

Chapter 18** 2. What needs to happen to avert?¶

3. Focus on demonstrable new conditions that

4. 16 equal portions likely push limits

5. Predicts largest well; not too far on small- 4. Obligations of reality?

est 5. Some things are out of our control

4. Gaping disparities on opposite poles is no 7. What type of activity tends to consume a

random fluke lot of power?

6. Brilliant future if you can figure out effective 8. Duty cycle

ways 9. Proportional to $\Delta T$

7. How else will change happen? (but elaborate… ) 10. Gasoline is about 4 times the other two

8. Will contribute 2–3% of the annual total 11. Just a bit less than average in all categories

9. A bit over half the global energy budget! 12. Close to twice the gas is used in the form of electricity

10. Two approaches: cynical or hopeful; make either pitch 13. Both in the same neighborhood

11. In the hundreds 14. Big disparity; which is more likely?

12. I was hoping you had some ideas 15. S.U.V. might not make the cut, but smaller cars will

13. Surprisingly far: almost two-thirds of the

Chapter 19¶

way

17. Is six-sevenths a coincidence?

18. Still could be a parasite, even if larger than a flea 18. As if one day a week is all dairy/eggs

19. Easier to break than make 19. Is it directed or emergent?

20. What things are dependent on growth to 20. Think frivolous or huge resource de-

operate normally? mand

5. What limits? 21. Do your best: might prevent the worst

6. Does it bear on humanity in some way? 22. Can you even tell the needle isn’t at full?