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Essay Instructions: - There is a limit of 2 websites as reference. I would prefer as many periodicals, articles, journals and magazines as possible.

- The paper should be technology based, no policies and procedures. Basically, what happens in the world.

-Reference list should be alphabetized

- If possible a short description on the history of RADIUS SERVERS and (I THINK), it will be replace by DIAMETER Servers, but I'm not sure.

The class is a WIRELESS NETWORK/SECURITY class, if your could find someway to apply it to wireless security, I'd appreciate it. Thanks in advance.

Essay Instructions: SOURCES HAS BEEN UPLOADED.

First off, have you read How to Do Activities, How to Write Up Activities (5-Point Form) and How to Submit Activities? If not, read these first or you may lose points.

SYNOPSIS: You are going to perform a simple observation similar to one performed more than 2000 years ago that led to the first reasonable determination of the circumference of the Earth. An ancient Greek-Egyptian astronomer by the name of Eratosthenes knew that on the first day of summer, in the city of Syenne some distance to the South of him, at local noon rays of the Sun fell to the bottom of a well. This indicated that the Sun was directly overhead. Eratosthenes also knew that on the same day, the rays of the Sun would not fall to the bottom of a well in his home of Alexandria, and hence the Sun was NOT directly overhead there. In fact, shadows from tall objects such as a tower indicated that the Sun was a certain degree (Ø) from being straight overhead.

Eratosthenes assumed that the Earth was a sphere. (Surprisingly to most modern Americans, many ancient Greeks believed that the Earth was round!) As such, he knew that geometrically, the angle (Ø) that he observed for the Sun (from the vertical) was he same as the angle between Alexandria, the center of the Earth, and Syenne.

As such, the ratio of the angle Ø to the full 360 degrees of a full circle is the same as the ratio of the distance between Alexandria and Syenne to the full circumference of the Earth. Eratosthenes believed the distance between Alexandria and Syenne was about 5000 stadia, and he measured the angle to be about 7 degrees. Thus,

(5000 stadia)/Circumference = 7 degrees / 360 degrees

or

Circumference = 500*360 stadia / 7 = 25714 stadia (approximately)

Unfortunately, we do not know exactly how far a stadium length was, and it does not matter for our activity, but given our best estimate, Eratosthenes measurement of the circumference was very good. In any event the technique is valid.

YOUR PURPOSE: Your purpose is the replicate, as well you can , the basic observations that Eratosthenes made, and to determine the size of the Earth. For our measurement, we'll use a shadow in Denver to determine the height of the Sun, and a hypothetical observer in New Mexico, 518 kilometers due South of Denver, will provide the other measurement. The difference between the two angles (Denver and New Mexico) will give us the angle between Denver, the center of the Earth, and the New Mexico site. This corresponds to the angle Ø in the graphic above. Then if we know the straight line, North-South distance between Denver and the New Mexico site (518 kilometers), we can in turn obtain the circumference of the Earth.


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THE PROCEDURE

1) Get an appropriate ruler or measuring device, preferably Metric, and learn how to use it. (See Mathhelp)
2) Determine Local Solar Noon for the date in question (See When to do it below)
3) Find or erect a gnomon (a "shadow stick") on a smooth, flat, level surface. Use nothing less than 12 inches or 305 mm.
4) Accurately measure the height of your gnomon above the ground or surface it stands on.
5) Measure the length of the shadow at local solar noon
6) Determine the angle of the Sun (use trig if you can, the javascript calculator if you can't)
6) Calculate the circumference and radius of the Earth
7) Answer the questions below
8) Write up the Activity using the 5-Point Format

When to Do It You need to have a sunny day and the exact time of Local Solar Noon. This is when the Sun is due South in the sky. It also is known as "Transit Time," and is what was called "High Noon" in olden times. It is not the same as clocktime Noon, from which it can deviate by as much as about 20 minutes before or after (even more when we have Daylight Time).

You can find Local Solar Noon by finding the time that is exactly half-way between sunrise and sunset. You should be able to find those times in the newspaper. Or you could just take the easy way out and go to this web page: calendar. Click on the "Rise/Set Times" button and it will call up the information from the U.S. Naval Observatory computer. On the resulting page, "Local Solar Noon" is called "Sun Transit."

This is for the Denver area only. If you live farther from Denver than say, Boulder or Castlerock, then you need to use the old-fashioned way by figuring the midpoint between sun rise and set times at your location. (If you live in another major metropolitan area, you can use the Naval Observatory site to figure it for you.) Also, if you live anywhere outside the immediate Denver metropolitan area, you need to contact me for an adjustment, since this it set up to do between Denver and the New Mexico location. If you live in New York or Chicago or San Francisco or Nome, or anywhere outside the Denver metro area -- you need to let me know because you will have to make an adjustment.

What to Do The basic idea of what you need to do is shown in the graphic to the right. You can find an existing setup (a fencepost or telephone pole, for example), or you can devise your own set up, or click here for a suggestion: alternate setup graphic). It is absolutely vital that whatever you use as a gnomon be straight and that it be perfectly erect (does not lean to one side or another). If it is not straight, or it leans even a degree or two, your observation and your result will be off significantly. Do not use a tree, and be careful if you use a fencepost or telephone pole, because they may appear straight and erect at first glance, but may not be on closer inspection.

If you choose to make your own gnomon, you can set this up in many different ways. For the gnomon, you will need a dowel or some other straight object. Strictly speaking, the taller the gnomon is, the better. However, it is difficult to assure that very short gnomons are straight and upright, so don't use anything less than about 305 mm (roughly a foot). It is very important that it be straight, and standing perfectly upright. Even if it is leaning only just slightly, or is even slightly bent or irregular by just a few millimeters, your results will be significantly in error. Take your measurements in millimeters (mm). Most school rulers today have a metric scale with millimeters and centimeters. If you don't know how to measure in millimeters, the math help page has some information. (See "Measurements: Metric and Imperial")

The important thing is that you get an accurate measurement of the exact height of your "shadow stick". The surface must be very flat, and the gnomon or shadow stick must be straight and not tilted. Its height is measured from the point where the shadow begins, to the top directly above that point. The shadow is measured from the stick where the shadow begins to the end of the shadow. If the shadow stick is thick, do not add the thickness of the stick to the length of the shadow.

Measurement & Calculations Be sure you have an accurate watch so that you make your measurement within a couple of minutes of Local Solar Noon. Then simply measure the length of the shadow, as accurately as you can. I cannot overemphasize the need for a flat, level measuring surface and accuracy in your measurement. If things are not set up properly, or your measurement is off by more than 4 or 5 mm, your results will be significantly off.

OK, now that you have taken your measurement, what does it mean? The length of the shadow is a function of how high the Sun was in the sky at the time. Local Solar Noon is the time when the Sun is highest on that particular date. The shorter the shadow, the higher the Sun, and the longer the shadow, the lower the Sun. We get short shadows in Summer, long shadows in Winter.

Click on this link to get to our special javascript calculator: Javacalc. You will compare your measurement with a similar measurement made for an observer in New Mexico, about 60 miles due East of Albuquerque. With this difference in angle, plus the distance from Denver to the New Mexico site (518 km), you can figure the total circumference of Earth via Eratosthenes' method. Note that Shadowcalc requires javascript capability. While most browsers should handle this with ease, if your browser chokes, contact me. (If you prefer to use trigonometry, go here: trigonometry, but it is not required.)

Given the value you have obtained for the circumference of the Earth, what is the radius of the Earth? Circumference equals two times Pi times the radius. So the radius is the Circumference divided by two times Pi. (For our purposes, Pi is 3.1416). Example:

If you measure the circumference as 30,000 km, then the radius is 30,000 km divided by 2 pi. R = 30,000 km/(2 X 3.1416) = 4,775 km (rounded). Note that this is not real data, just an example to show the calculation.

Check your results against your textbook(s), some other reference, or through a simple online search. If you fail to check your results (when it is easily possible), you may lose points. I am not looking for extreme precision, but if your result is not within about 10 percent of the accepted value, then you did something wrong. If you don't understand something, ask.

Questions: These must be answered, or you will lose points. In your write up, answer these after your statement of conclusions. List each question separately as shown below, followed on the next line by your answer. Do not jumble all the questions and answers together in a paragaph, or you will lose points.
1) Why did you have to do this at local solar noon, that is, when the Sun is due South in the sky?
2) If you had done this significantly too early or too late, what kind of error would there have been your shadow length?
3) If you had done this significantly too early or too late, what kind of error would there have been your circumference value?
4) Could astronauts on the Moon or Mars use this technique to determine the circumference of those objects?

Finally, write up your activity Online students use the submission form. Lecture class students should format your activity according to the standard 5-point plan, which includes a header with your name, date, activity, class and so on; a paragraph explaining your purpose; a paragraph explaining the procedure; a statement or table of the data you took and calculations made; and a statement of conclusions. Note that the statement of conclusions does not mean your results. Instead, tell me what you learned from the activity, not just the results you obtained. Include the Questions above in list form, followed immediately by your answers. Remember, always check your results.

In your statement of Data, Observations and Results, include all the important data (the time of Local Solar Noon, the gnomon height, the shadow length you measured, the angle you obtained, the location of your observation (city is good enough), and the angle for New Mexico). Part of this project is to determine how aware you are of what is important and what is not, so please give some thought to this. Don't just throw things in "for completeness." Ask yourself, what would someone else need to know to repeat this observation? What is relevant and what is not? You could lose points for not including relevant data, and just as surely you can lose points for adding irrelevant detail. Of course also state your specific results for circumference and radius.

Questions: These must be answered, or you will lose points. In your write up, answer these after your statement of conclusions. List each question separately as shown below, followed on the next line by your answer. Do not jumble all the questions and answers together in a paragaph, or you will lose points.
1) Why did you have to do this at local solar noon, that is, when the Sun is due South in the sky?
2) If you had done this significantly too early or too late, what kind of error would there have been your shadow length?
3) If you had done this significantly too early or too late, what kind of error would there have been your circumference value?
4) Could astronauts on the Moon or Mars use this technique to determine the circumference of those objects?

I encourage students to work together, but you should take your own measurements and write up your own report.

NOTE: This activity is very sensitive to errors in measurement, probably more so than any other activity. This means that a small error in measurement can mean a large error in your results. You must be as careful as possible to ensure that the gnomon is standing straight, and that the surface on which you measure is very flat and level. Your measurement should be as accurate as possible, to within a couple of millimeters.


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Essay Instructions: 1. Find the radian measure of the central angle of a circle of radius r = 4 inches that intercepts an arc length s = 20 inches.
2. In which quadrant will the angel 100 degrees lie in the standard position?
3.
4. In which quadrant will the angle -305 degrees lie in the standard position?
5. Find the length of the arc on a circle of radius r = 5 yards intercepted by a central angle 0 = 70 degrees.
6.
7. Convert the following angle to degrees: n or pie
5 5
8. Classify the angle 101 degrees as acute, right, obtuse, or straight.
9. Draw the following angle in standard the position: 7n or 7pie
4 4
10. Convert -60 degrees to radians. Express the answer as a multiple of pie.
11.
12. Find a co-terminal angle for the following angle: -268 degrees
13. Find the value of (sin 38 degrees) (csc 38 degrees)
14. Use an identity to find the value of: sin^2 50 degrees + cos^2 50 degrees

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Customer is requesting that (mangway) completes this order.

Essay Instructions: SYNOPSIS
Reflect sunlight from a small mirror onto a flat, light-colored wall or other flat surface as directed. Measure the distance from the mirror to the image on the wall; and measure the diameter of the image. Plug the two measurements into a simple formula to calculate the diameter of the Sun.

BACKGROUND
The Sun's distance – and hence its diameter – was not known accurately until the famous British naval Captain James Cook and others observed the 1769 transit of Venus. With views from various locations, astronomers could determine the parallax and hence the absolute distance to Venus. Knowing the relative distance to the planets from Kepler's Laws, they could then work out the distance to the Sun. There was some discrepancy in the results from various observers, but a reasonably accurate distance to the Sun (about 150 million kilometers) was derived from the data in 1835 by the astronomer Encke. Here the radius of the orbit, which we now know averages about 150 million kilometers, is designated by the letter "L." Knowing the radius of the orbit and by measuring the apparent size of the sun in the sky, astronomers could easily determine the diameter of the sun using trigonometry or the simple geometry of this activity.


NOTE:It is perfectly safe to look at the image on the wall, but do not look directly at the Sun or into the reflection from the mirror. The diagram above is just one possibility -- you don't have to do it exactly like this. You can place the mirror just outside the window or even on a windowsill where the Sun is shining. Or you can stand in a driveway and reflect it into a garage, or just set it up outside and reflect onto the wall of a building in the shade.(To see the set up another student used, click here: student example)


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In the diagram above, the distance from the sun to the mirror ( L ) bears the same relationship to the sun's diameter ( D ) as the distance from the mirror to the solar image ( l ) does to the diameter of the solar image ( d ). You know that L averages 150 million km. To find D , the diameter of the sun, you only need to set up a mirror reflection as indicated, and then measure l and d .

D/L=d/l
or
D=(d/l)*L
The slash (/) represents division, whereas the asterisk (*) means "multiplied by."

PROCEDURE
A flat compact mirror will work fine, but to keep the distance "l" down to a workable figure, you must stop the mirror down to a smaller size, perhaps 6-8 mm, by covering the mirror with paper in which a small hole has been punched. (Experiment with sizes and distances to find something that works for you. Also keep in mind that the ratio of the size of the reflection distance ("d") to the size of the mirror cutout should be in the order of 800 or 1000 to one. A 6 meter reflection from a 7 mm mirror fits this ratio. Do NOT use reflection distances ("l") of less than 5 meters or more than 7 meters. Be sure that whatever mirror you use is flat and does not magnify.)

You must figure out a way to hold the mirror steady so you or a friend can go measure the diameter of the reflected image. Make the hole in the cardboard (to show the mirror) 6 to 8 millimeters across, and then arrange the distance "l" so that it is about 5-7 meters. This is for practicality. The exact size and shape of the hole in the cardboard is not important, but it should be roughly 6-8 mm across, and the reflection distance ("l") should be 5 - 7 meters. The larger the size of the mirror (or cutout), the longer the reflection distance must be.

(Save yourself some trouble by reading the Example and Notes below before you start any measurements!)

You should reflect the light onto a light-colored surface, but one that is in an darkened area. This is so the image will show up better. Do not use a dark-colored surface or wall because it will make it hard to see. Use common sense.

With a mirror (cutout) of about 6-8 millimeters, a reflection throw ( l ) of about 6 meters should work. Keep in mind that the reflected image of the Sun will be faint, so it needs to be reflected into a darkened area. If your distance l is too short, the image will be brighter but the edges will be fuzzy. If l is too great, the image will be too faint to measure. If your cutout is too small, the image will be too faint. If it is too large, the edges of the reflected image will be fuzzy. Experiment to find an arrangement where the image is large enough, sharp enough, and bright enough to measure accurately.

Make at least 3 measurements both of the distance l and the diameter d . (Just measure the distance and diameter 3 times, trying to be as accurate as possible. You likely will come up with slightly different values each time.) Put the data in a table, and apply the formula above to determine the diameter of the sun. Then average your results. Be sure to take the average of your results -- do not average the measurements you take. Treat this as a regular lab activity, supplying all five components as usual. Consider and describe possible sources of error.

HOW TO MEASURE: It is difficult to measure the size of the reflection on a wall using a ruler. Instead, print out these two pages of circles:

(Don't just print out this entire page and use the graphic to the right. It will not be to the correct scale. You must click on the image to the right and then print out the resulting two-page PDF file. This is just a suggestion. You can directly measure the image on the wall if you like. If you don't have a PDF reader, there is a nice free one here: http://www.foxitsoftware.com/pdf/rd_intro.php)

If you follow the directions, then your reflection on the wall should closely fit one of the circles. (If it is just a bit larger than one circle and a bit smaller than another, you can interpolate). Find the closest match and then measure the diameter of the circle as your "d." Use a metric ruler with millimeters (mm). Measure "l" in meters and convert to mm (1 meter = 1000 mm) If the reflection on the wall is oval (it should be round) or is significantly larger or smaller than any of the circles provided, then your set up is wrong and needs to be fixed.

Check your results against your textbook(s), some other reference, or through a simple online search. If you fail to check your results (when it is easily possible), you may lose points. I am not looking for extreme precision, but if your result is not within about 10 percent of the accepted value, then you did something wrong. If you don't understand something, ask.

Here is a table to put your data and results in. Remember, do NOT average your data. Instead, figure out "D" for each set of measurements, and then average your results for "D." That will go in the green box and is your answer. (NOTE: The table example provided here is for lecture-class students. Online students do not have to use this, but instead will use the form in the online submission file.)

"d" (in mm) "l" (in mm) D=(d/l)*150,000,000 km
1
2
3
Average: (don't average data) (don't average data)

Remember that your "d" and "l" must be in the same units (e.g., millimeters or "mm")

By the way, you know that the distance to the sun varies through the year, ranging from about 146 million kilometers in early January, to about 151million kilometers in July. You could find out the exact distance on the day of your observation, but the level of accuracy of this observation doesn't warrant it. It is sufficient to use the figure of 150 million kilometers.

END

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EXAMPLE: Let's say that you make your measurements from 6 meters (6000 mm) and that one of your measurements is 64 mm. In this case, the "d" is 64 mm and the "l" is 6000 mm. The ratio (or equation!) becomes:
D=(64mm/6000mm)*L.

Note that "*" means to multiply. Since L is given to you as 150 million kilometers, the equation now becomes:

D=(64mm/6000mm)*150,000,000 km.

Now, note the "mm" in the "64mm/6000mm" part. The "mm" of course stands for "millimeters." Since you have them on both sides of the division symbol ("/"), you can and should simply remove them because they (the "mms" that is) cancel each other out leaving you with:

D=(64/6000)*150,000,000 km.

Now, 64 divided by 6000 equals 0.01067 (rounded). Some people mess themselves up by doing it the wrong way. Instead of dividing 64 by 6000, they divide 6000 by 64. NO! Read it the right way. On a calculator, you would enter "64/6000" like this:

64 ÷ 6000 = answer. Punch in "64" then the divide symbol (÷ or /), then "6000" followed by "="

So now you have got

D = 0.01067 * 150 million km.

Just multiply (*) the 150 million by 0.01067. On a calculator, punch in "150000000" (no commas) followed by the multiply symbol (usually "X" or "*" on a calculator) followed by "=" to get: 1,600,500 (rounded a bit). Putting in the commas you get: 1,600,500 km. Your answer would be 1,600,500 km.

Then repeat this 2 more times, average your results for the final answer.

Be sure to show all pertinent data and your calculations.

NOTE: This is an incorrect answer because the data I used (64 mm and 6000) were incorrect. I just used it for the example. Follow this procedure with your own measurements and you should have no trouble.

Always watch the units (meters, millimeters, etc.) you are working in. Don't confuse them and don't try to do something like divide millimeters by meters or multiply kilometers by meters. If you have a ratio, such as "A/B", then be sure that both A and B are in the same units (such as inches), and then the units will cancel out in the ratio to leave you a "pure" dimensionless number like 0.25 or 245 without any inches. Also note that there is no need whatsoever to change everything into miles as some students do. For instance, you do NOT need to change a measurement like 64 millimeters into miles. It just complicates things and makes it much more prone to mistakes.

And please don't confuse miles with kilometers. Your book usually uses kilometers (km) instead of miles. A mile is longer than a kilometer. It takes 1.609 kilometers to make 1 mile. If you get them confused your answer will be way off, so stay with millimeters, meters and kilometers.

Don't make life any harder than it is.

PRACTICAL NOTE : It will take a little practice to learn how to aim the mirror where you want it, and again, it is very convenient to have someone help you with this. I suggest that you try this handheld first to get a feel for it. After that find a way to aim the mirror and have it stay in place at least for a minute or two while you make your measurements. No one can hold it stead enough by hand to do this. It needs to be mounted firmly in some manner. I mounted the mirror on an old camera and mounted that on a tripod for easy aiming. You could try devising a mounting with a cardboard box, or by holding the mirror with a large lump of modeling clay. Be inventive.
The important points are that the mirror must be small compared to the distance of reflection ("l"), which should be about 6 meters, and that you must get an image bright enough to measure. The exact size of the mirror is not important as that measurement does not figure in. It is the ratio of the distance of reflection ("l") to the size of the reflected image ("d") that is important. The actual size of the mirror determines how easy or hard the image will be to see, but if your reflection distance is large enough, the size of the mirror does not affect the ratio. The projected (reflected) image should be round. If it is not round, your measurements will be wrong. An oblong image indicates a problem with your set up. If you need to, adjust your set up until you get a round image.

MATH NOTE : You should take your measurements in metric units -- that is, meters and millimeters. If you do not understand how to do this, check the measurement help page or some other reference: measurements. (There is also lots of information here: http://lamar.colostate.edu/~hillger/ )If you can't find a metric ruler or your tape measure does not have metric units, you will have to convert into metric units. One inch is 25.4 millimeters, and one foot is about 305 millimeters. In any event, your "d" and "l" must be in the same units.



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