# Assignment 6: Measure the distance to downtown Rochester

Due Friday, Apr 28, 2000, at the beginning of class

Your job this week is to use three different techniques to climb a "distance ladder" and determine the distance from the Imaging Sciences building to downtown Rochester.

The methods you will use are

1. pacing
2. using windows as "standard yardsticks"
3. using buildings as "standard yardsticks"

Each of these three steps introduces some uncertainty. You must estimate the uncertainty in each step, and the overall uncertainty in the distance to downtown Rochester.

You will need to go inside the Center for Imaging Sciences and up to its third floor to do part of this assignment. Make sure that you go there while the building is unlocked.

#### Step I: Pace to Building 7B

Start at the main entrance to Building 76, on its eastern side. There's an asphalt walkway to Building 7B, leading to the "tunnel" through it. If you walk along the path, and continue past the "tunnel" without going into it, you'll come to some shrubbery. Go around the shrubbery to the north face of Building 7B. There is a row of windows, which open into rooms below the ground level. The seventh window along the wall, away from the "tunnel", should have a bird feeder standing on its sill.

You must measure the distance from the front door of Building 76 to the seventh window of building 7B, the one with the bird feeder. Use your pace to measure the distance: figure out how far each of your paces is, then use your paces as you go from Building 76 to Building 7B. This might not be so easy. You can walk straight from Building 76 to the "tunnel", but you can't go straight through the shrubbery; you need to find some way to estimate the final segment of the distance, from the "tunnel" to the seventh window.

Write down the distance (in meters), and your estimate for the uncertainty in the distance, in terms of percent (that is, you might write 53 meters, plus or minus 5 percent).

#### Step II: Use windows to go from Building 7B to Building 50

The tallest dormitory on the RIT campus is Building 50, aka Ellingson Hall. You next job is to determine the distance from Building 76 to Building 50, without going there. Here's how:

1. make the assumption that all windows on all buildings are the same size. This isn't true, but it might not be terribly wrong.
2. walk into the Building 76 (the Center for Imaging Science), up go up the stairway to the third floor. Walk to the North-East corner of the third-floor corridor (it will be the first corner to the left as you leave the stairs on the third floor)
3. stand in the very North-East corner of this area, so that you can look down and to the right and see Building 7B. You should just barely be able to see the "tunnel" from this vantage point.
4. look at the windows on the ground floor of Building 7B; the seventh from the tunnel should have the bird feeder you saw earlier.
5. find some way to measure the apparent angular size of the seventh window: you could use your fingers, or a pencil, or some other device as a reference
6. now look due East, across campus. Building 50 is the tallest dorm in that area. Examine it carefully, so that you can discern the windows. Again, find some way to measure the apparent angular size of the windows. It might be difficult, because they appear so small, due to their distance
7. calculate the ratio of sizes:
```              angular size of seventh window in Building 7B
R1 =  ----------------------------------------------
angular size of windows in Building 50
```
8. this is also the ratio of the distances of the buildings:
```        dist to Bldg 50  =   R1 * (distance to Building 7B)
```

Calculate the ratio R1, and also estimate an uncertainty in its value, in percent. Write down both the ratio and its uncertainty. Use your ratio to calculate the distance to Bldg 50, in meters.

In this step, you are using windows as standard yardsticks: objects which all have exactly the same size. They don't, in real life, so you calculation will be wrong; but at least it's a reasonable estimate.

#### Step III: Use buildings to go from RIT campus to downtown Rochester

In this step, you will assume that all big buildings are the same size, and use them as standard yardsticks. Again, it's not true, but it's better than nothing. Here's what to do:

1. again, go up to the North-East corner of the corridor on the third floor of Building 76.
2. look at Building 50, the tall dorm across campus. Make a measurement of its apparent angular size.
3. now look to the North-East. In the distance, just over the horizon, you should see several of the tallest buildings in downtown Rochester. Pick the building just to the right of the tallest skyscraper. Measure its apparent angular size.
4. calculate the ratio
```                  apparent angular size of Building 50
R2   =  --------------------------------------------
apparent angular size of downtown skyscraper
```
and write it down
5. estimate the uncertainty in this ratio, in percent. Write it down, too
6. now, if all these two buildings are the same height (and they probably aren't), then this ratio will be the same as the ratio of their distances. So you can use the ratio R2 to calculate the distance to downtown:
```     dist to downtown  =  R2  *  (distance to Building 50)
```
Calculate the distance to downtown Rochester, in meters.

Once again, you are making a somewhat dubious assumption: all big buildings are the same height. Once again, it's likely to be wrong --- but perhaps it's not wrong by too much.

So, what is your distance to downtown Rochester? Is it reasonable? Do you think it's correct?

In each step, there was some uncertainty. To figure out the overall uncertainty in the distance to Rochester, you need to add up the percentage uncertainty in each step:

```    total percent uncertainty  =  (percent uncertainty in step I)  +
(percent uncertainty in step II) +
(percent uncertainty in step III)
```
How large is your total uncertainty?

You can find the uncertainty in meters (instead of percent) by multiplying the percentage uncertainty by the distance to Rochester. So, for example, if you calculated

```       distance          =   5000   meters
total uncertainty   =     25   percent

uncertainty         =   (5000 meters) * (0.25)
=    1250  meters

--->  distance        =    5000  +/-  1250  meters

```
Write down the uncertainty in your estimate, in meters.

Finally, use any device -- a map, a car, a GPS unit -- to find the REAL distance between the RIT campus and downtown Rochester. Compare that to your estimate. Was your value equal to the real distance, within your uncertainty?