To find the length of side \( a \) of triangle \( \triangle ABC \), we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2}ab \sin C \]
Where \( a \) and \( b \) are the lengths of two sides and \( C \) is the angle between them.
From the information provided:
- Area = 134 square feet
- \( b = 24 \) feet
- \( m\angle C = 41^\circ \)
We can substitute these values into the area formula and solve for \( a \):
\[ 134 = \frac{1}{2} \cdot a \cdot 24 \cdot \sin(41^\circ) \]
First, we calculate \( \sin(41^\circ) \):
\[ \sin(41^\circ) \approx 0.6561 \]
Now we substitute this value into the equation:
\[ 134 = \frac{1}{2} \cdot a \cdot 24 \cdot 0.6561 \]
Simplifying we get:
\[ 134 = 12a \cdot 0.6561 \]
\[ 134 = 7.8732a \]
Now, to find \( a \):
\[ a = \frac{134}{7.8732} \approx 17.0 \]
Rounding to the nearest whole number, we find:
\[ a \approx 17 \text{ feet} \]
Thus, the length of side \( a \) is 17 feet.
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