To find the length of side \( a \) in triangle \( ABC \) given the area of the triangle, angle \( C \), and side \( b \), we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2}ab \sin(C) \]
Where:
- \( a \) is the length of side \( a \)
- \( b \) is the length of side \( b \) (given as 16 km)
- \( C \) is the angle \( C \) (given as 73 degrees)
- The area of the triangle is given as 92.5 km²
Substituting the known values into the area formula gives:
\[ 92.5 = \frac{1}{2} \times a \times 16 \times \sin(73^\circ) \]
First, calculate \( \sin(73^\circ) \):
\[ \sin(73^\circ) \approx 0.9563 \]
Now substitute this value into the equation:
\[ 92.5 = \frac{1}{2} \times a \times 16 \times 0.9563 \]
Simplifying further:
\[ 92.5 = 8a \times 0.9563 \]
Multiply the right side:
\[ 92.5 = 7.6504a \]
To isolate \( a \), divide both sides by 7.6504:
\[ a = \frac{92.5}{7.6504} \approx 12.1 \]
Rounding to the nearest km, we get:
\[ a \approx 12 \text{ km} \]
Thus, the length of \( a \) is \( \boxed{12} \) km.
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