Question

During a half-moon phase, the Earth-Moon-Sun form a right triangle. If the Moon-Earth-Sun angle at this half-moon phase is measured to be \(89.85^\circ\), the ratio of the Earth-Sun and Earth-Moon distances is closest to:

• A. 328
• B. 382
• C. 238
• D. 283

Hint:

For problems involving celestial distances and angles, trigonometric functions such as tangent are useful to determine ratios based on given angular measurements.

Answer 1

The Correct Option is B

Step 1: Given the right triangle formed by Earth, Moon, and Sun during the half-moon phase, the Moon-Earth-Sun angle is provided as \(89.85^\circ\). 

Step 2: Using trigonometry in a right triangle: \[ \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \] Here, the opposite side is the Earth-Sun distance, and the adjacent side is the Earth-Moon distance. Given that the Moon-Earth-Sun angle is \(89.85^\circ\), the tangent of the angle can be approximated using: \[ \tan(89.85^\circ) \approx 382 \] 

Step 3: Therefore, the ratio of Earth-Sun to Earth-Moon distances is approximately 382. 

Conclusion: Based on the calculation, the correct answer is option (B) 382.