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
• E.

Hint:

For problems involving right triangles and trigonometric ratios, ensure accurate angle measurements and use precise trigonometric values for calculations.

Answer 1

The Correct Option is B

Step 1: Use the trigonometric relationship.
During the half-moon phase, the Earth-Moon-Sun form a right triangle. Using the tangent of the Moon-Earth-Sun angle \( 89.85^\circ \): \[ \text{tan} \, \theta = \frac{\text{Earth-Moon distance}}{\text{Earth-Sun distance}}. \]
Rewriting: \[ \text{Earth-Sun distance} = \frac{\text{Earth-Moon distance}}{\text{tan} \, \theta}. \]
Step 2: Substitute the values.
With \( \theta = 89.85^\circ \), \( \text{tan} \, \theta \approx 0.002618 \). Hence:
\[ \frac{\text{Earth-Sun distance}}{\text{Earth-Moon distance}} = \frac{1}{0.002618} \approx 382. \]
Step 3: Conclusion.
The ratio of Earth-Sun to Earth-Moon distances is closest to \( 382 \).