The Greek mathematician Eratosthenes (ca. 276-195 B.C.) measured the circumference of the earth from the following observations. He noticed that on a certain day the sun shone directly down a deep well in Syene (modern Aswan). At the same time in Alexandria, 500 miles north (on the same meridian), the rays of the sun shone at an angle of 7.2° to the zenith. Use this information and the figure to find the radius of the earth. (Round your answer to the nearest ten miles.)


Use this information and the figure to find the circumference of the earth.

I don't know what your diagram looks like it, but here is page that is similar to your problem

http://www.sciencebuddies.org/science-fair-projects/project_ideas/Astro_p018.shtml

notice that we have a central angle of 7.2°
we can set up a ratio
x/500 = 360°/7.2°
x = 25 000

So the circumference of the earth is 25000 miles according to his calculations.

(modern calculations show the circumference at the equator to be 24902 miles.)

The radius would then be 3,979 miles

Well, Eratosthenes really took it to great lengths to measure the circumference of the earth! He must have had a lot of time on his hands, huh? Anyway, let's get to the calculation.

First, we need to find the angle between the rays of the sun and the vertical line in Syene. Since the rays were shining directly down the well, we can assume that the angle is 90 degrees. Now, in Alexandria, the angle was 7.2 degrees to the zenith.

Next, we can use some trigonometry to find the distance between Syene and Alexandria. Since the earth is basically a super huge sphere, we can consider the arc between these two cities as a fraction of the whole circumference of the earth.

Using the formula: Circumference = 2 * pi * radius, we can rearrange it to solve for the radius: Radius = Circumference / (2 * pi).

Now, we know that the distance between Syene and Alexandria is 500 miles, and that it represents an arc on the earth's surface. We can set up a proportion to find the total circumference: 500 miles is to the total circumference as 7.2 degrees is to 360 degrees.

Using this proportion, we can find the total circumference of the earth. And voila, we have the radius!

I guess Eratosthenes really nailed the idea of "thinking outside the box" to measure the earth. His method might sound a bit crazy, but hey, it worked!

To find the radius of the Earth using the given information, we can make use of a few geometric principles. Here's how we can go about it:

Step 1: Calculate the distance between Syene and Alexandria using the given information. We are told that the distance between the two cities is 500 miles.

Step 2: Determine the central angle formed at the center of the Earth by drawing a line between Syene, the center of the Earth, and Alexandria. Since the Earth is a sphere, the central angle formed at the center will be the same as the angle formed by the sun's rays in Alexandria (7.2°).

Step 3: Understand that the central angle formed at the center of a circle is proportional to the length of the arc it cuts out on the circle's circumference. In this case, the arc represents the distance between Syene and Alexandria (500 miles).

Step 4: Calculate the measure of a full circle's central angle. Since a full circle has 360°, we can set up a proportion to find the measure of the central angle corresponding to the full circumference of the Earth. The proportion will be:
7.2° / 360° = 500 miles / x, where x represents the full circumference of the Earth.

Step 5: Solve for x, which is the circumference of the Earth. Cross-multiply and then divide by 7.2° to find x.

Step 6: Once you have the circumference of the Earth, you can calculate the radius by dividing the circumference by 2π (since the circumference of a circle is equal to 2π times the radius).

Step 7: Finally, round the radius to the nearest ten miles.

By following these steps, you will be able to find the radius of the Earth.