Math Problem Statement

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.)

Solution

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 SEM=89.85\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=240,000EM = 240,000 miles.

We need to find the distance ESES 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:

    • EE as the vertex at the right angle.
    • EM=240,000EM = 240,000 miles (the distance from the Earth to the Moon).
    • SEM=89.85\angle SEM = 89.85^\circ.

    The triangle SEMSEM has:

    • Hypotenuse ESES (the distance from Earth to the Sun).
    • Opposite side EM=240,000EM = 240,000 miles.
  2. Use Trigonometric Functions: We can use the sine function: sin(SEM)=opposite sidehypotenuse=EMES\sin(\angle SEM) = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{EM}{ES} Substituting the given values: sin(89.85)=240,000ES\sin(89.85^\circ) = \frac{240,000}{ES}

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

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

Let's compute ESES.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 SEM\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.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangles
Sine Function

Formulas

sin(θ) = opposite / hypotenuse
ES = EM / sin(θ)

Theorems

Trigonometric Ratios in Right Triangles

Suitable Grade Level

Grades 9-12