The length of the shadow of a vertical tower is $15$ m when the altitude of the sun is $60^\circ$. What will be the length of the shadow when the altitude of the sun becomes $45^\circ$?
Options:
- A: $15$ m
- B: $15\sqrt{3}$ m
- C: $\frac{15}{\sqrt{3}}$ m
- D: $30$ m
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