To find the length of side \( a \) in triangle \( ABC \) given the area, side \( b \), and angle \( C \), we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \]
Here, we know:
- Area = 134 square feet
- \( b = 24 \) feet
- \( \angle C = 41^\circ \)
Plugging the known values into the area formula, we can solve for \( a \):
\[ 134 = \frac{1}{2} \times a \times 24 \times \sin(41^\circ) \]
Now, calculate \(\sin(41^\circ)\):
\[ \sin(41^\circ) \approx 0.6561 \]
Substituting this value into the equation gives:
\[ 134 = \frac{1}{2} \times a \times 24 \times 0.6561 \]
This simplifies to:
\[ 134 = 12 \times a \times 0.6561 \]
Calculating \( 12 \times 0.6561 \):
\[ 12 \times 0.6561 \approx 7.9932 \]
Now, we can substitute this back into the equation:
\[ 134 = 7.9932 \times a \]
To solve for \( a \):
\[ a = \frac{134}{7.9932} \approx 16.76 \]
Rounding this to the nearest whole number, we find:
\[ a \approx 17 \]
Therefore, the length of side \( a \) is \( \boxed{17} \) feet.
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