ES 10 - Homework 2
The Age of the Universe
Greg Anderson

Due 16 January 1998

This is the first real homework set for ES 10, worth a total of 90 points. You may find Handout 1 to be useful, especially on the last question. Please write your answers on paper other than this assignment, make sure your name is on each page, and staple your pages together.

If you have problems or questions, please come to the discussion section, or see me after class, or send me an e-mail. My e-mail address is anderson@python.ucsd.edu.

Question 1. Write the following numbers as powers of ten. (4 points)

1. 1000
2. 1 000 000
3. 0.001
4. 1

Question 2. Write the following numbers in ``normal'' notation. (4 points)

1. 10³
2. 10⁶
3. 10^-4
4. 10^-2

Question 3. Do the following problems. (4 points)

1. log(10⁴) = ?
2. Given that log(x) = 6, what is x?
3. 10⁵ × 10^-3
4. 10^-6 ÷ 10^-8

Question 4. Write the following numbers in scientific notation. (4 points)

1. 123
2. 345 000
3. 0.006453
4. 0.3421

Question 5. Write the following numbers in ``normal'' notation. (4 points)

1. 1.23 × 10³
2. 0.25 × 10^-4
3. 3.45 × 10⁵
4. 2.22 × 10⁸

Question 6. Perform the following calculations. Show your work for full credit. (4 points)

1. (2.0 × 10⁴) × (3 × 10^-7)
2. (4.2 × 10^-3) × (0.14 × 10³)
3. (6.85 × 10^-6) ÷ (6.85 × 10^-9)
4. (2.55 × 10⁵) ÷ (5.05 × 10^-10) × (1.10 × 10^-8)

Question 7. Show your work for full credit. (8 points)

A cop pulls you over for speeding on a road trip to Las Vegas. He says, ``I clocked you at 4470 cm/sec. The speed limit here is 31.3 m/sec. I'll have to arrest you for reckless driving. You can get out when you make bail.'' If bail is $20 per mile/hour over the speed limit, how much are your friends going to have to pay to get you out of jail? Round your answer to the nearest dollar.

Question 8. Show your work for full credit. (8 points)

You've decided to throw a big Super Bowl party to root for the San Francisco 49ers. You've decided to get a keg of Samuel Adams to serve at your party.

1. Given that a keg holds about 15.3 gallons, how many 16-ounce beers can you get from the keg? Round your answer to the nearest beer.
2. Given the answer to part a, how much will each 16-ounce beer cost if the keg costs $105? Round your answer to the nearest cent.

Question 9. The Age of the Universe (50 points)

In this problem, you will calculate the age of the universe from a set of numbers which Dr. Tauxe has put together (see Table 1). You will use these numbers to make a graph, and then you will use your graph to estimate the age of the universe.

Here's what I want you to do:

1. Since the age of the universe in seconds is a really big number, convert the numbers in Table 1 into kilometers (km) for distance and kilometers/year (km/yr) for speed. You will want to know that there are 365.2422 days in a year.
2. Make a table of the numbers you computed in the previous step. Put the speed (in km/yr) in the first column and distance (in km) in the second column.
3. Make a graph of the numbers in your table. Put speed on the X-axis and distance on the Y-axis.
4. Draw a line on your graph which looks like the best line to describe the slope of your cloud of points. Don't play connect-the-dots; draw a line through your points. By eyeball is fine, and if you know some more fancy way of doing it, that's fine too.
5. Using your graph with the straight line (and your optional reading of the hint for this problem), estimate the age of the universe. If you do your work correctly, your estimate for the age will have the units of time.

Here is what I want you to turn in for this problem:

1. Your table of speed (in km/yr) and distance (in km), based on the numbers in Table 1.
2. Your graph of the points in your table. My rules for graphs:
   • Neatness counts - for a lot. Please make your graph as neat and clean as possible.
   • Don't plot your graph on regular binder paper with hand-drawn ``graph-paper'' lines. If you have access to a computer with some graphing software, it is really quite easy to make nice graphs quickly. If you don't have access to a computer with graphing software on it, please use graph paper.
   • Add an informative label to both the X- and Y-axes, something like ``Distance (in km)'' or ``Speed (in km/yr)''. A title is nice, too, but not necessary.

   Please follow these rules. It makes life nicer for both of us.

3. On your graph, please write the slope of the line you have drawn. Use scientific notation and state your units.
4. A neat summary of the math you did in order to solve the problem, like how you converted the numbers in Table 1 to the numbers in your table, and how you calculated the slope of the line. Anything else you want me to see on how you did the math, please put in this summary.
5. On your summary sheet, also please write the slope of the line you drew. Please don't forget to use scientific notation and to state your units.

Table 1:  Retreat Speeds of and Distances to Distant Galaxies

Speed (cm/sec)	Distance (cm)
9.683 × 107	6.918 × 1025
1.109 × 108	7.079 × 1025
1.452 × 108	8.128 × 1025
1.603 × 108	7.762 × 1025
3.819 × 108	2.239 × 1026
5.152 × 108	2.512 × 1026
5.370 × 108	2.818 × 1026
4.581 × 108	3.020 × 1026
5.058 × 108	3.467 × 1026
6.653 × 108	3.090 × 1026
6.516 × 108	3.715 × 1026
6.252 × 108	4.169 × 1026
8.730 × 108	4.169 × 1026
8.356 × 108	5.248 × 1026
1.012 × 109	5.248 × 1026
1.021 × 109	6.166 × 1026
1.343 × 109	6.607 × 1026
1.352 × 109	8.128 × 1026
1.618 × 109	8.128 × 1026
1.667 × 109	9.772 × 1026
2.328 × 109	1.000 × 1027
2.535 × 109	1.288 × 1027
2.173 × 109	1.259 × 1027
2.228 × 109	1.479 × 1027
2.495 × 109	1.514 × 1027
4.887 × 109	1.778 × 1027
4.498 × 109	2.455 × 1027
4.093 × 109	2.570 × 1027
5.420 × 109	3.020 × 1027
6.699 × 109	2.692 × 1027
7.047 × 109	3.020 × 1027
7.621 × 109	3.890 × 1027
6.209 × 109	3.715 × 1027
7.047 × 109	4.677 × 1027
1.122 × 1010	5.248 × 1027
1.349 × 1010	5.012 × 1027

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