To find the length of side \( b \) (opposite angle \( B \)) in triangle \( ABC \) where \( A = 30^\circ \), \( B = 45^\circ \), and side \( a = 10 \) units, we can use the Law of Sines, which states:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
First, we need to find \( \sin A \) and \( \sin B \):
- \( \sin(30^\circ) = \frac{1}{2} \)
- \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \)
Now applying the Law of Sines:
\[ \frac{10}{\frac{1}{2}} = \frac{b}{\frac{\sqrt{2}}{2}} \]
This simplifies to:
\[ 20 = \frac{b}{\frac{\sqrt{2}}{2}} \]
Multiplying both sides by \( \frac{\sqrt{2}}{2} \):
\[ b = 20 \cdot \frac{\sqrt{2}}{2} = 10\sqrt{2} \]
Approximating \( 10\sqrt{2} \):
\[ 10\sqrt{2} \approx 10 \cdot 1.414 \approx 14.14 \]
Since this falls between the provided responses, we check if it matches any of the response options by estimating \( 10\sqrt{2} \).
The answer closest to \( 14.14 \) which we computed is not among the options, which suggest checking further into an approximation or calculation of a side. However, for which \( b \) relative to the options of 10.0, 7.07, 11.18, and 15.0 might have shifted.
However, in case of rounded outputs,
The remaining alternatives misalign however: Therefore, The Law of Sines simplifies yields no exact from choices provided. It indicates meaningful solutions may match only those listed if provided both were confirmed in error to compare precisely by unit accuracy / yielding.
Calculating thus means output does not yield select better result within trajectory given choice options!
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