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The man's name was Eratosthenes, and the city was Syene, a town on the Nile river, which Eratosthenes knew to be (the equivalent of) 496 miles from the bustling city of Alexandria. At some point it struck Eratosthenes that he could use this information to get a good approximation of the radius of the earth. (Do you know the radius of the earth?) At noon of summer solstice, he calculated the angle of the sun's rays in Alexandria by using a vertical stick, measuring its shadow, and applying basic trigonometry (of right triangles). He found that the sun's rays fell 7 degrees and 12 minutes from vertical. Now imagine the earth as a large circle with the cities of Syene and Alexandria lying on the circle. They can be represented by two points separated by a distance of 496 miles. The 496 miles is the measure of the arc length from one point to the other on the circle. Go ahead and sketch this picture with Syene directly at the top of the circle and Alexandria a little to the right of Syene. Now draw a line from the center of the circle straight through Syene and another from the center through Alexandria. The line through Syene points directly to the sun at noon of summer solstice. The line through Alexandria represents the stick Eratosthenes used to calculate the sun's rays. This stick and its shadow form two sides of a right triangle whose hypotenuse is parallel to the sun's rays. The shadow of the stick forms the base of this right triangle. We know the top angle of the triangle is 7 degrees and 12 minutes based on the calculation of Eratosthenes.

The man's name was Eratosthenes, and the city was Syene, a town on the Nile river, which Eratosthenes knew to be (the equivalent of) 496 miles from the bustling city of Alexandria. At some point it struck Eratosthenes that he could use this information to get a good approximation of the radius of the earth. (Do you know the radius of the earth?) At noon of summer solstice, he calculated the angle of the sun's rays in Alexandria by using a vertical stick, measuring its shadow, and applying basic trigonometry (of right triangles). He found that the sun's rays fell 7 degrees and 12 minutes from vertical. Now imagine the earth as a large circle with the cities of Syene and Alexandria lying on the circle. They can be represented by two points separated by a distance of 496 miles. The 496 miles is the measure of the arc length from one point to the other on the circle. Go ahead and sketch this picture with Syene directly at the top of the circle and Alexandria a little to the right of Syene. Now draw a line from the center of the circle straight through Syene and another from the center through Alexandria. The line through Syene points directly to the sun at noon of summer solstice. The line through Alexandria represents the stick Eratosthenes used to calculate the sun's rays. This stick and its shadow form two sides of a right triangle whose hypotenuse is parallel to the sun's rays. The shadow of the stick forms the base of this right triangle. We know the top angle of the triangle is 7 degrees and 12 minutes based on the calculation of Eratosthenes.

Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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The man's name was Eratosthenes, and the city was Syene, a town on the Nile River, which Eratosthenes knew to be (the equivalent of) 496 miles from the bustling city of Alexandria. At some point it struck Eratosthenes that he could use this information to get a good approximation of the radius of the earth. (Do you know the radius of the earth?) At noon of summer solstice, he calculated the angle of the sun's rays in Alexandria by using a vertical stick, measuring its shadow, and applying basic trigonometry (of right triangles). He found that the sun's rays fell 7 degrees and 12 minutes from vertical. Now imagine the earth as a large circle with the cities of Syene and Alexandria lying on the circle. They can be represented by two points separated by a distance of 496 miles. The 496 miles is the measure of the arc length from one point to the other on the circle. Go ahead and sketch this picture with Syene directly at the top of the circle and Alexandria a little to the right of Syene. Now draw a line from the center of the circle straight through Syene and another from the center through Alexandria. The line through Syene points directly to the sun at noon of summer solstice. The line through Alexandria represents the stick Eratosthenes used to calculate the sun's rays. This stick and its shadow form two sides of a right triangle whose hypotenuse is parallel to the sun's rays. The shadow of the stick forms the base of this right triangle. We know the top angle of the triangle is 7 degrees and 12 minutes based on the calculation of Eratosthenes. Now use our arc length formula to get an approximation of the radius of the earth. (Hint: What is the angle formed at the center of the circle by the two lines that pass through Syene and Alexandria?)
Transcribed Image Text:The man's name was Eratosthenes, and the city was Syene, a town on the Nile River, which Eratosthenes knew to be (the equivalent of) 496 miles from the bustling city of Alexandria. At some point it struck Eratosthenes that he could use this information to get a good approximation of the radius of the earth. (Do you know the radius of the earth?) At noon of summer solstice, he calculated the angle of the sun's rays in Alexandria by using a vertical stick, measuring its shadow, and applying basic trigonometry (of right triangles). He found that the sun's rays fell 7 degrees and 12 minutes from vertical. Now imagine the earth as a large circle with the cities of Syene and Alexandria lying on the circle. They can be represented by two points separated by a distance of 496 miles. The 496 miles is the measure of the arc length from one point to the other on the circle. Go ahead and sketch this picture with Syene directly at the top of the circle and Alexandria a little to the right of Syene. Now draw a line from the center of the circle straight through Syene and another from the center through Alexandria. The line through Syene points directly to the sun at noon of summer solstice. The line through Alexandria represents the stick Eratosthenes used to calculate the sun's rays. This stick and its shadow form two sides of a right triangle whose hypotenuse is parallel to the sun's rays. The shadow of the stick forms the base of this right triangle. We know the top angle of the triangle is 7 degrees and 12 minutes based on the calculation of Eratosthenes. Now use our arc length formula to get an approximation of the radius of the earth. (Hint: What is the angle formed at the center of the circle by the two lines that pass through Syene and Alexandria?)
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