To find the length of side \( a \) in triangle \( ABC \), we can use the formula for the area of a triangle involving two sides and the included angle:
\[ \text{Area} = \frac{1}{2} \times b \times c \times \sin(A) \]
However, since we need to find side \( a \), we first use the formula for the area in terms of the sides \( a \), \( b \), \( c \) and the angle \( C \):
\[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \]
Given the values:
- Area = 134 square feet
- \( b = 24 \) feet
- \( C = 41^\circ \)
We can substitute these values into the area formula:
\[ 134 = \frac{1}{2} \times a \times 24 \times \sin(41^\circ) \]
Now, we can solve for \( a \):
- First, calculate \( \sin(41^\circ) \):
\[ \sin(41^\circ) \approx 0.6561 \quad (\text{using a calculator}) \]
- Substitute this back into the equation:
\[ 134 = \frac{1}{2} \times a \times 24 \times 0.6561 \]
- Simplify the equation:
\[ 134 = 12a \times 0.6561 \] \[ 134 = 7.8732a \]
- Now, solve for \( a \):
\[ a = \frac{134}{7.8732} \approx 17.0 \]
Rounding this to the nearest whole number gives:
\[ a \approx 17 \text{ feet} \]
Thus, the length of side \( a \) is approximately \( \boxed{17} \) feet.
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