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 94,500,000 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.)

Verified step by step guidance

1. Identify the similar triangles involved in the problem: one triangle formed by the sun and its shadow, and the other by the moon and its shadow.
2. Set up the proportion using the diameters and distances: \( \frac{\text{Diameter of Sun}}{\text{Distance from Sun to Earth}} = \frac{\text{Diameter of Moon}}{\text{Distance from Moon to Earth}} \).
3. Substitute the given values into the proportion: \( \frac{865,000}{94,500,000} = \frac{2,159}{x} \), where \( x \) is the maximum distance from the Moon to Earth.
4. Solve the proportion for \( x \) by cross-multiplying and isolating \( x \): \( 865,000 \times x = 2,159 \times 94,500,000 \).
5. Divide both sides by 865,000 to find \( x \), the maximum distance from the Moon to Earth for a total solar eclipse.

Key Concepts

Similar Triangles

Similar triangles are triangles that have the same shape but may differ in size. This concept is crucial for solving problems involving proportional relationships, such as those found in geometry and trigonometry. In the context of the solar eclipse, the relationship between the distances and sizes of the sun, moon, and Earth can be analyzed using similar triangles to determine the maximum distance the moon can be from Earth while still casting a shadow.

Proportional Relationships

Proportional relationships occur when two quantities maintain a constant ratio. In trigonometry, this concept is often applied to solve problems involving angles and lengths in similar triangles. For the solar eclipse problem, understanding how the diameters of the sun and moon relate to their distances from Earth allows for the calculation of the maximum distance the moon can be positioned while still achieving a total eclipse.

Umbra and Penumbra

The umbra is the darkest part of a shadow, where the light source is completely blocked, while the penumbra is the lighter outer part where the light is partially obscured. In the context of a solar eclipse, the umbra is the area on Earth where a total eclipse occurs. Understanding the distinction between these two types of shadows is essential for determining the conditions under which a total solar eclipse can happen, particularly in relation to the distances involved.