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¶

4. Between 250 and 300 years

5. Between 250 and 300 years

6. Between 10¹⁸ and 10²⁰

7. Between 100 and 150

8. More than 10⁴⁰

9. Between 5 and 500

10. Later than 23:50

11. Before 12:10 AM

12. Between 10 and 40 years

13. Between 75 and 100 years

14. Between 300 and 500 billion

15. It’s not 100 times longer

16. A few millennia

17. Between 50 and 100 W

18. A smidge higher than boiling

19. Between 200 and 250 K

20. Between 150 and 250 K

21. Between 100 and 275 K

Chapter 2¶

1. Nearly $100 billion

2. 4%: $\sim$$1 trillion; 5%: more than $100 trillion

3. On the low end of advanced countries

4. Between 25 and 100 MJ/$

5. Between 5% and 50%

6. The text had trading art, singing lessons, therapy, and financial planning

7. Between 100 and 1,000; Less than 10 to go

8. Between 20 and 100 years

9. May help to think of something once prevalent, now rare

10. Especially fruitful might be biological dependencies

11. It can’t all be free of material substance

Chapter 3¶

1. Less than a third

2. Comparable to U.S. population today

3. Comparable to world population 200 years ago

4. Table 3.2 offers a rough check

5. Over 16 billion; less than half the time we now experience

6. Pretty close to Table 3.2 except for first two entries

7. Answer must be less than 14 billion; whereas Problem 6 was in excess of 15 billion

8. Two are negative; three are positive

9. Between 1 and 5%

10. Add almost a half million; more than half million born; less than half million died

11. Answers should round to the table values

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

13. Correct results are in the table

14. See Figure 3.15

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

16. It nearly triples

17. Area is key

18. Lesotho is relevant

Chapter 4¶

1. Earth: smaller than peppercorn and basketball-court distant; Moon: sand grain a hand’s width away

2. Comparable to the actual Earth radius

3. 1 AU = 1 km; Earth 1/12,000 km

4. A fast walk or slow jog

5. Think about the subtended angle

6. Multiply sets to get accumulated scale factors

7. A good deal farther than the moon, but still well short of the sun/Mars

8. Ratio is more than a billion, and would take more than 4 lifetimes

9. Think in terms of area as fraction of plot space

10. Text has climbing Mt. Everest, supersonic commercial flight, squirrel obstacle course, and economic decoupling

11. Will take 15–20 tanks of gas, and achieve a fuel economy a factor of 30 or so below typical cars

12. Double the gasoline from previous problem; gasoline mass almost as much as the car itself

Chapter 5¶

1. Several inches

2. About the length of a typical room

3. A few kJ total, most in sliding

4. You’ve got this

5. Nearly 1 GJ

6. Two of the points in Example 5.2.1 offer guidance

7. Figure 5.1 offers hints

8. Roughly half human metabolic power

9. Less than 5 seconds

10. Between 50–100 kcal (200–400 kJ)

11. Sensibly, a little less than 2 minutes

12. Results should be roughly consistent with Figure 5.2

13. A bit less than 10% of household electricity

14. They’re actually close, within 10%

15. More than two

16. Several kWh; less than $1

17. On the low end of the human metabolism range; the equivalent cost of 10–20 burritos

18. A few hundred W

19. Comparable to running a clothes dryer (Fig. 5.2)

20. Several hundred MJ

21. Over 100 kWh; 2–4 burritos-worth

22. Several Therms; cost of fast-food lunch

23. Several gallons; cost of fast-food lunch for two

24. One is about twice the other

25. A little in excess of 10 kW

26. Less than a quarter of estimated

27. Between 1 and 2 hours per day

28. The largest number is near 10⁸

29. Six of the entries are inverses of six others

30. A little over an Amp

31. Nearly 10 kW; will cost over $1,000; don’t do this!

32. Will last 2–3 hours

33. In line with most chemical reactions, in the 50–200 kJ/mol range

34. Between 3–$5\times 10^{-19}$ J per photon; get more than $10^{18}/$sec

35. In the neighborhood of 1 $\mu \mathrm{m}$ or 1 eV

Chapter 6¶

1. Approx. 200 kJ, depending on mass

2. Several minutes

3. Several minutes

4. About 5 minutes

5. A few hours

6. Not below freezing

7. Not quite up to “room” temperature

8. Not quite half the time

9. The cost of two burritos per day

10. Instances of heat/flame causing movement

11. See Table 6.2

12. Roughly 30 kJ and 100 J/K

13. Between 5–10%

14. A couple dozen percent, roughly

15. Pushing 100%, but not quite there

16. Achieves about 1/3 of theoretical

17. $\Delta T > 50^{\circ}\mathrm{C}$; environment not that cold

18. Close to a dozen kJ

19. Twice, twice

20. Just short of 5 years

Chapter 7¶

1. a) between 30–40%; b) almost all; c) close to 2/3; d) roughly a quarter

2. Coal is near 12 qBtu, for instance

3. Nuclear is about 22%, for instance

4. Residential is about 5 qBtu, for instance

5. Industry is a little over 30%, for instance

6. About 14% is renewable, for instance

7. Less than 10%

8. Between 5 and 10%

9. It is one of the fossil fuels

10. Well over 100 years

11. Surprisingly soon: maybe before student loans paid off

12. Nothing to see here

13. Nothing to see here

14. Pay attention to the dashed line

15. Pay attention to the dashed line

Chapter 8¶

1. All lines overlap the up-slope

2. Likely vs. hopeful?

3. Many features unchanged

4. Won’t be zero into future

5. What enabled, then disappeared?

6. Opposite of ideal

7. Did not behave like U.S.

8. Based on energy density

Chapter 9¶

1. A single integer works okay for all three

2. Nearly 100 kg

3. Between 10–15 kg

4. Approaching 1 GJ, and human-mass scale

5. Total is like small adult or large child

6. More than a factor of two

7. Get about 50 years; rate not constant

8. The numbers basically match

9. Between 1–2 ppm$_{\mathrm{v}}$, in agreement with Figure 9.3

10. What is it we know?

11. Seems deserving of high marks

12. Historical vs. current activity levels

13. About $10^{\circ}\mathrm{C}$ cooler than actual

14. Two pure cases and one partial

15. Several degrees warmer

16. Very good for us at the right level

17. Numbers are not far from realistic

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

19. End $\sim 3^{\circ}\mathrm{C}$ high; almost linear, but not quite

20. No need to balance: Nature doesn’t bother

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

29. Sum to about 700 years; almost all in ice and ocean

30. A few hundred meters

31. A finger’s breadth per year

32. Keen to hear your thoughts

33. Keen to hear your thoughts

Chapter 10¶

1. Mostly clean; not all, though

2. Nothing is free

3. What would unlimited mean?

4. Table 10.2 has some help

5. Can’t rely on any sun-driven energy

6. Between 200–250 $\mathrm{W/m}^{2}$

7. Photosynthesis supports essentially all life

8. Comparing numbers in TW

9. More than half

10. A little less than 1%

11. Not far from 1,000 $\mathrm{W/m}^{2}$

12. Nearly 10 degrees

13. Look for crazy-big input

14. Between 0.5–1 gallon

15. More than $4,000\times$

Chapter 11¶

1. Roughly 20 kJ

2. About 10 stories of a building

3. Close to 0.1 kJ

4. About 4 times higher than airliners travel

5. About two-thirds Earth radius

6. Try using half the mass and half the energy

7. Cube is roughly as big as height from ground

8. About 6 times typical nuclear plant

9. Nearly 200 m

10. A little shy of 500 $\mathrm{m}^{3}/\mathrm{s}$

11. Between 50–75%

12. Roughly 50%

13. About a million homes

14. Approaching 10,000 cubic meters per second

15. You’ve got a little over an hour

16. Less than 1 TW in the end

17. Between 1–2 meters

Chapter 12¶

1. A few Joules

2. Roughly $1^{\circ}\mathrm{C}$

3. Something like 10 m/s

4. Mass shows up in both $mgh$ and $\frac{1}{2}mv^{2}$

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

8. Around about 8 times

9. Follow the cube…

10. Runs approximately 10 kW to 1 MW

11. Roughly two-thirds the original speed

12. Close to 10 MW

13. Closer to 10 m/s than to 15 m/s

14. Almost double freeway speeds

15. Between 5–10 m/s

16. In the ballpark of 70 kW

17. Recover 0.65%

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

19. Outer box area corresponds to running at 100%, full time

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 needs

Chapter 13¶

1. How big are the packages?

2. Something times 10²¹

3. Roughly 10¹⁶

4. About 1,000

5. About 4,000 times

6. Use Eq. 13.3 to guide your reasoning

7. Should match Figure 13.1

8. One micron for each finger?

9. Think about spill-over into UV and/or IR

10. Peak around $2.5 \times 10^{8}$, about 1 $\mu \mathrm{m}$ wide; matches up well

11. Think energetics and depth

12. Is the answer transparent?

13. Just comparing two energies

14. Several hundred km/s

15. Condense the saga to that of a winner

16. Answer might involve physics, biology, rooftops

17. Inversely: larger in one means smaller in the other

18. Already extremely similar

19. Think of current as a rate of electron flow in the circuit

20. Get very close to 1,360 $\mathrm{W/m}^{2}$

21. Sweltering is not preferred

22. Between 5–6 $\mathrm{kWh/m}^{2}/$day; between 200–250 $\mathrm{W/m}^{2}$

23. Involves interpreting $\mathrm{kWh/m}^{2}/$day as full-sun-hours

24. Not far from 200 $\mathrm{W/m}^{2}$

25. Range straddles 200 $\mathrm{W/m}^{2}$, varying about 10%

26. Best at latitude; almost 15% better than flat

27. Approaches 6 $\mathrm{kWh/m}^{2}/$day

28. Large house (and just the PV for one person)

29. Square is about as wide as Arizona or California east-to-west

30. Cost, surely—but other challenges and mismatches as well

31. A little over 200 W

32. Roughly the size of a bedroom

33. Will spend a little over $4,000

34. A little over a decade

35. Even lower than $\sim 20\%$ from insolation vs. overhead

36. About $2-worth of sun

37. Hint: study Figures 13.23 and 13.24

38. In absolute terms…

Chapter 14¶

1. About a dozen tons of CO$_{2}$

2. Between 0.1–0.5%

3. Almost 100 logs per person per year

4. In the neighborhood of half-dozen logs per day

5. Won’t be exactly 15 years, but close

6. Almost 1.5 L of ethanol

7. Roughly consistent with Table 14.1 for coal

8. Net is one-third production

9. Extra land is twice the yield-land

10. A bit longer than a U.S. Presidential term

11. Nothing to spare

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

13. Box barely fits north–south in U.S.

14. Personal preferences play a role

Chapter 15¶

1. A blueberry

2. $N = 8$

3. Use $Z = 26$ to get there

4. Should match quite well

5. Two diagonals have no gray squares

6. One has a half-life longer than a million years

7. Roughly twice as old as agriculture

8. Between 1–2%

9. One is about 3% of the other (both decay)

10. Step right

11. That last step might take a while

12. Two decays do it

13. Sand does the job

14. Somewhere between a car and a bus?

15. Close to 1 kg

16. Around a couple-dozen micrograms

17. Table should match

18. Energy has a mass, via $E = mc^{2}$

19. Adds about 1% to the mass

20. Not far from 1,500 MeV

21. Not much

22. Figure 15.14 is relevant

23. It’s a strontium isotope

24. $A$ is twice a prime number

25. Stick to 80 $< A < 110$ and 125 $< A < 155$ to respect distributions

26. Mid-20s of MeV

27. From steam onwards, it’s basically the same

28. Between 3 and 5 cents per kWh

29. A few per week!

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

31. Almost 2 million tons

32. A few hundred tons

33. Less than a decade

34. Two stand out

35. Centuries

36. More often than once every two years

37. A nearly exact match!

38. Worked out in text: no calculation necessary—just interpretation

39. Energy jump size

40. Like a milk jug

Chapter 16¶

1. Shortfall is more than a factor of 200

2. A bit farther than the record

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

4. Ranges about 55–85%

5. Geothermal is a bit less than 1% of alternative electricity

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

7. Works out

8. Diameter like a small house’s footprint

9. Comparable to human metabolism; 1% of American demand

10. A little more than 6 times that in Example 16.4.1

Chapter 17¶

3. Algae who?

4. Two words almost say it all

5. Fine if it is a little shy: transfer rates vary

6. Think about what a house can access, and steam plants

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

Chapter 18¶

1. 16 equal portions

2. Predicts largest well; not too far on smallest

3. Gaping disparities on opposite poles is no random fluke

4. Brilliant future if you can figure out effective ways

5. How else will change happen? (but elaborate…)

6. Will contribute 2–3% of the annual total

7. A bit over half the global energy budget!

8. Two approaches: cynical or hopeful; make either pitch

9. In the hundreds

10. I was hoping you had some ideas

Chapter 19¶

1. Still could be a parasite, even if larger than a flea

2. Easier to break than make

3. What things are dependent on growth to operate normally?

4. What limits?

5. Does it bear on humanity in some way?

6. Wait; who has my…

7. Focus on what has mattered until now

8. What’s the alternative?

9. Please figure out how it can work!

10. Is this the movie version, or the real-life one?

Chapter 20¶

2. What needs to happen to avert?

3. Focus on demonstrable new conditions that likely push limits

4. Obligations of reality?

5. Some things are out of our control

6. What type of activity tends to consume a lot of power?

7. Duty cycle

8. Proportional to $\Delta T$

9. Gasoline is about 4 times the other two

10. Just a bit less than average in all categories

11. Close to twice the gas is used in the form of electricity

12. Both in the same neighborhood

13. Big disparity; which is more likely?

14. S.U.V. might not make the cut, but smaller cars will

15. Surprisingly far: almost two-thirds of the way

16. Is six-sevenths a coincidence?

17. As if one day a week is all dairy/eggs

18. Is it directed or emergent?

19. Think frivolous or huge resource demand

20. Do your best: might prevent the worst

21. Can you even tell the needle isn’t at full?