Question

The earth is approximately 92.9 million miles from the sun, and 240,000 miles from the moon. When the angle sun-earth-moon is 25°, how far is the moon from the sun?

Answer 1

Answer on Question #77646 – Math – Trigonometry

Question

The Earth is approximately 92.9 million miles from the Sun, and 240,000 miles from the Moon. When the angle Sun-Earth-Moon is $25{}^{\circ}$, how far is the Moon from the Sun?

Solution

Let’s set the Earth’s position as point $E$, the Sun’s position as point $S$ and the Moon’s position as point $M$. Then we have triangle $SEM$ as illustrated in the Figure, where

$$|SE| = 9.29 \cdot 10^7 \text{ miles};$$ $$|ME| = 2.4 \cdot 10^5 \text{ miles};$$ $$\alpha = 25{}^{\circ}.$$

According to the law of cosines:

$$|MS|^2 = |SE|^2 + |ME|^2 - 2|SE||ME| \cos \alpha,$$ $$|MS|^2 = (9.29 \cdot 10^7)^2 + (2.4 \cdot 10^5)^2 - 2 \cdot 9.29 \cdot 10^7 \cdot 2.4 \cdot 10^5 \cdot \cos 25{}^{\circ}.$$

Concerning the fact that $\cos 25{}^{\circ} = 0.9063$, we have:

$$|MS|^2 = (9.29 \cdot 10^7)^2 + (2.4 \cdot 10^5)^2 - 2 \cdot 9.29 \cdot 10^7 \cdot 2.4 \cdot 10^5 \cdot 0.9063$$ $$|MS|^2 = 86.3 \cdot 10^{14} + 5.76 \cdot 10^{10} - 44.59 \cdot 10^{12}$$ $$|MS|^2 = 863000 \cdot 10^{10} + 5.76 \cdot 10^{10} - 4459 \cdot 10^{10}$$ $$|MS|^2 = 858546.76 \cdot 10^{10}$$ $$|MS| = 926.6 \cdot 10^5 = 9.27 \cdot 10^7 \text{ miles}.$$

So the Moon is 92.7 million miles from the Sun.

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