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The distance from the Sun to Earth is approximately $\"\"$ .

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The distance from the Sun to Mercury is approximately $\"\"$ .

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The elongation angle is the angle formed between the line of sight from Earth to the Sun and the line of sight from Earth to Mercury $\"\"$ .

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Consider the distance between the earth and mercury as $\"\"$ .

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Draw a diagram for the situation:

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$\"\"$

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Observe the figure:

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$\"\"$ .

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Apply the Law of Sines for triangle $\"\"$ :

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ and $\"\"$ .

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$\"\"$ .

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$\"\"$ .

\

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Apply the Law of Sines for triangle $\"\"$ for $\"\"$ :

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$\"\"$

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$\"\"$

\

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Apply the Law of Sines for triangle $\"\"$ for $\"\"$ .

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$\"\"$

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$\"\"$

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Thus, Possible distances between Earth and Mercury are approximately $\"\"$ or $\"\"$ .

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Possible distances between Earth and Mercury are approximately $\"\"$ or $\"\"$ .