Question

Solve each problem. Solar Eclipse on Earth The sun has a diameter of about 865,000 mi with a maximum distance from Earth's surface of about 94,5​​00,000 mi. The moon has a smaller diameter of 2159 mi. For a total solar eclipse to occur, the moon must pass between Earth and the sun. The moon must also be close enough to Earth for the moon's umbra (shadow) to reach the surface of Earth. (Data from Karttunen, H., P. Kröger, H. Oja, M. Putannen, and K. Donners, Editors, Fundamental Astronomy, Fourth Edition, Springer-Verlag.) a. Calculate the maximum distance, to the nearest thousand miles, that the moon can be from Earth and still have a total solar eclipse occur. (Hint: Use similar triangles.)

Accepted Answer

Identify the similar triangles formed by the sun, moon, and their shadows. The sun and moon create two similar triangles: one with the sun's diameter and distance from Earth, and the other with the moon's diameter and the maximum distance from Earth where the moon's shadow just reaches Earth. Set up the proportion using the diameters and distances. Let $$D_s = 865,000$$ miles be the sun's diameter, $$d_s = 94,500,000$$ miles be the sun's distance from Earth, $$D_m = 2,159$$ miles be the moon's diameter, and $$d_m be $$the maximum distance from Earth to the moon for a total eclipse. The proportion is:

$$\frac{D_s}{d_s} = \frac{D_m}{d_m}$$ Rearrange the proportion to solve for $$d_m$$:

$$d_m = \frac{D_m 	imes d_s}{D_s}$$ Calculate $$d_m$$ using the given values to find the maximum distance the moon can be from Earth for a total solar eclipse. Remember to keep the units consistent and round the final answer to the nearest thousand miles. Interpret the result: this distance represents the farthest the moon can be from Earth while still casting a shadow that reaches Earth's surface, allowing a total solar eclipse to occur.

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Concetti chiave

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Proportional Reasoning in Astronomy