HW 2 - The Age of the Universe and Other Bugaboos

Due 17 January 1997

This is the first real homework set for ES 10. It consists of 6 quickie-type
questions each worth 5 points, two longer questions each worth 10 points, and
one big question, worth 50 points. The big question will require you to use
several of the subjects discussed in Handout 1, which you should also have.
This homework assignment is **due at the beginning of lecture on 17 January.**

If you have problems or questions, please come see me in my office hours or send me an e-mail. My office hours are Monday and Wednesday right after class, from 2:00-3:30 pm, in 1539 Galbraith Hall. I will not hold office hours on 20 January, due to the long Martin Luther King, Jr. holiday. I will try to arrange extra office hours during the week of 20 January to make up for my absence. If you can't make my office hours, or you have a question outside of them, please feel free to send me an e-mail. We can either try to answer the problem via e-mail or we can try to arrange an appointment either down here at SIO, or up on the main campus. My e-mail address is ganderson@ucsd.edu

**Question 1.** Write the following numbers as powers of ten. (5 points)

- 100
- 1,000,000
- 0.000001
- 10
- 0.0001

**Question 2.** Write the following numbers in ``normal'' notation. (5 points)

**Question 3.** Do the following problems. (5 points)

- Given that , what is x?
- Given that , what is x?

**Question 4.** Write the following numbers in scientific notation. (5 points)

- 123,000
- 345,000,000,000,000
- 0.00000000678
- 0.005921
- 0.5091

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

- 5.0 E 03

**Question 6.** Perform the following calculations. **Show your work
for credit.** (5 points)

**Question 7. Show your work for full credit.** (10 points)

A cop pulls you over for speeding. He says, ``I clocked your speed at 33.5
m/s. The speed limit here is 29.1 m/s. I'll have to give you a ticket.'' If
tickets cost $5 per mph over the speed limit, how much is the fine for this
ticket?

**Question 8. Show your work for full credit.** (10 points)

You've decided to throw a big Super Bowl party to root for the Carolina
Panthers. You've decided to get a keg of Sierra Nevada Pale Ale to serve at
your party.

- Given that a keg holds about 15 gallons, how many 12-ounce beers can you get from the keg?
- Given the answer to the previous question, how much will each 12-ounce beer cost if the keg costs $113? 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 I'll give you. I hope it will be fairly interesting, and not just
drudgery for you. First, a little background physics will probably be helpful.

Let's say I drive up to Magic Mountain next month to ride some roller coasters. It's roughly 180 miles away up north of LA, for those who don't know. If I drive a nice steady 60 mph (no doubt getting yelled at by everyone else on the road), how long would it take me to get there? You know the answer almost immediately, no doubt, but let's look at it in equation terms.

The relationship between distance, speed, and time is given by

From Equation 2, you will probably see that you can answer my first
question easily. Since , all you have to do is divide the
distance by the speed, and you get the answer, which in my case is 3 hours.
By the way, you probably also noticed that Equation 2 is the equation
for a straight line. That means that the relationship between distance and
speed is a *linear* one, and that if you plot a graph with distance as
a function of speed, you will see a straight line.

OK, now the problem. Dr. Tauxe has put together a table (Table 1) which gives distances to and speeds of a bunch of distant galaxies. You will use these numbers to make a graph, draw a line on that graph, and thus find the age of the universe.

Actually, if you're sharp, you'll notice that Table 1 doesn't
actually contain the distances and speeds, but their *logarithms*. So you
will have to convert these logs back into the original numbers in order to make
your graph.

So here's what I want you to do:

- Convert the logarithms in Table 1 to the original numbers to which the logarithms correspond.
- Those numbers are in centimeters (cm) for the distance and centimeters/second (cm/s) for the speed. Since the age of the universe in seconds is a really big number, convert the numbers you got in the previous step into kilometers (km) for distance and kilometers/year (km/yr) for speed. To do this, you will need to figure out how to convert cm to km, and how to convert seconds to years. For both of these, look at Handout 1. You will also want to know that there are 365.2422 days in a year.
- 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.
- Make a graph of the numbers in your table. Put speed on the X-axis, and distance on the Y-axis. Your graph should look basically like a straight line would go through your points fairly nicely.
- 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**By eyeball is fine, and if you know some more fancy way of doing it, that's fine too.*through*your points! - Figure out the slope of that line.
- That slope is your guess at the age of the universe, and if you did all the previous steps correctly, your guess should have units which make sense for an age.

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

- Your table of speed (in km/yr) and distance (in km), based on the numbers in the table below.
- 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. To that end:
- Plot your graph on graph paper, or using a computer. If you have access to a Macintosh with Cricket Graph or some other 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. If you don't have any graph paper, please buy some. If you don't want to or can't afford to buy graph paper, I've attached a sheet of graph paper to the back of this homework. Please use it.
- 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.

- On your graph, please indicate the slope of the line you have drawn. Use scientific notation and state your units.
- A neat summary of the math you did in order to solve the problem, like how to convert cm/s to km/yr, 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.
- On your summary sheet, also please indicate the slope of the line you drew, since this is your estimate of the age of the universe. Please don't forget to use scientific notation and to state your units.

The table of logarithms, Table 1, is shown below.

**Table 1:** Logarithms of Retreat Speeds of and Distances to Distant Galaxies

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Greg Anderson

ganderson@ucsd.edu

Tue Jan 14 12:14:49 PST 1997