To find the length of side \( c \) in triangle \( AABC \) with sides \( a = 17 \) ft, \( b = 20 \) ft, and angle \( C = 19^\circ \), we can use the Law of Cosines. The Law of Cosines states:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
Substituting the known values into the formula:
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Calculate \( a^2 \): \[ a^2 = 17^2 = 289 \]
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Calculate \( b^2 \): \[ b^2 = 20^2 = 400 \]
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Calculate \( 2ab \): \[ 2ab = 2 \cdot 17 \cdot 20 = 680 \]
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Calculate \( \cos(19^\circ) \): Using a calculator (make sure it's in degree mode), \[ \cos(19^\circ) \approx 0.9455 \]
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Now, substitute these values into the Law of Cosines formula: \[ c^2 = 289 + 400 - 680 \cdot 0.9455 \] \[ c^2 = 689 - 642.34 \approx 46.66 \]
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Finally, take the square root to find \( c \): \[ c \approx \sqrt{46.66} \approx 6.83 \]
Rounding to the nearest foot, we find:
\[ c \approx 7 \text{ ft.} \]
Thus, the length of side \( c \) is 7 ft.