When the moon is exactly half full, the earth, moon, and sun form a right angle (see the figure). At that time the angle formed by the sun, earth, and moon (using the earth as its vertex) is measured to be 89.85°. If the distance from the earth to the moon is 240,000 mi, estimate the distance from the earth to the sun. (Round your answer to one decimal place.)

To solve this problem, we will use trigonometry to estimate the distance from the Earth to the Sun when the Moon is half full.

### Problem Breakdown

Given:
1. The angle \(\angle \text{SEM} = 89.85^\circ\), where S is the Sun, E is the Earth, and M is the Moon.
2. The distance from the Earth to the Moon, \(EM = 240,000\) miles.

We need to find the distance \(ES\) from the Earth to the Sun.

### Step-by-Step Solution

1. **Model the Situation Using a Right Triangle**:
   When the Moon is half full, the Earth, Moon, and Sun form a right triangle with:
   - \(E\) as the vertex at the right angle.
   - \(EM = 240,000\) miles (the distance from the Earth to the Moon).
   - \(\angle SEM = 89.85^\circ\).

   The triangle \(SEM\) has:
   - Hypotenuse \(ES\) (the distance from Earth to the Sun).
   - Opposite side \(EM = 240,000\) miles.

2. **Use Trigonometric Functions**:
   We can use the sine function:
   \[
   \sin(\angle SEM) = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{EM}{ES}
   \]
   Substituting the given values:
   \[
   \sin(89.85^\circ) = \frac{240,000}{ES}
   \]

3. **Solve for \(ES\)**:
   Rearranging the equation for \(ES\):
   \[
   ES = \frac{240,000}{\sin(89.85^\circ)}
   \]

   Now, calculate the value of \(\sin(89.85^\circ)\).

Let's compute \(ES\).The estimated distance from the Earth to the Sun is approximately **240,000.8 miles**.

Given that this value is almost the same as the distance from the Earth to the Moon (240,000 miles), it shows that the angle \(\angle SEM\) is very close to 90 degrees, making the two distances nearly the same in this calculation.

Would you like more details on this solution or have any questions?

Here are 5 related questions to explore further:
1. What is the significance of the Moon being "half full" in this context?
2. How would the calculation change if the angle were slightly different?
3. Why does the small angle difference lead to such a large distance in astronomy?
4. How can we measure the distances in space more accurately?
5. What are some other ways to estimate the distance between celestial bodies?

**Tip**: In trigonometry, small changes in angles can lead to significant differences in distances, especially in astronomical contexts.

This problem demonstrates how to estimate the distance from the Earth to the Sun when the Moon is half full using trigonometric functions. The problem provides key information about the angle between the Earth, Sun, and Moon (89.85°) and the distance from Earth to the Moon (240,000 miles). Suitable for students in Grades 9-12.

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