To find the length of side \( a \) in triangle \( ABC \) using the given height \( h \) and angle \( m∠C \), we can use the sine of the angle formula, which relates the height and angle in a right triangle.
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Recognize that side \( a \) is opposite angle \( C \), the height \( h \) (which is 10 meters) corresponds to \( b \) in this case, and we can express side \( a \) in terms of the height and angle:
\[ h = a \cdot \sin(m∠C) \]
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Plug in the known values: \[ 10 = a \cdot \sin(39°) \]
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Solve for \( a \): \[ a = \frac{10}{\sin(39°)} \]
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Calculate \( \sin(39°) \): \[ \sin(39°) \approx 0.6293 \]
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Substitute back: \[ a = \frac{10}{0.6293} \approx 15.9 \]
Since we need to round to the nearest whole number: \[ a \approx 16 \text{ meters} \]
Thus, the length of side \( a \) is approximately \( 16 \) meters.
Final Answer: 16 meters
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