Question

a triangular swimming pool measures 42 ft on one side and 32.6 ft on another side. the two sides form an angle that measures 40.9°. how long is the third side? the length of the third side is □ ft. (simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.)

Explanation:

Step1: Identify the Law of Cosines formula

To find the length of the third side \( c \) of a triangle when two sides \( a = 42 \), \( b = 32.6 \) and the included angle \( C = 40.9^\circ \) are known, we use the Law of Cosines: \( c^2=a^2 + b^2-2ab\cos(C) \)

Step2: Substitute the given values

Substitute \( a = 42 \), \( b = 32.6 \), and \( C = 40.9^\circ \) into the formula:
\( c^2=42^2+32.6^2 - 2\times42\times32.6\times\cos(40.9^\circ) \)
First, calculate \( 42^2 = 1764 \), \( 32.6^2=1062.76 \)
Then, calculate \( 2\times42\times32.6 = 2733.6 \)
And \( \cos(40.9^\circ)\approx0.756 \)
So \( c^2=1764 + 1062.76-2733.6\times0.756 \)
\( c^2=1764 + 1062.76 - 2066.6016 \)
\( c^2=2826.76 - 2066.6016 \)
\( c^2 = 760.1584 \)

Step3: Take the square root

Take the square root of \( c^2 \) to find \( c \):
\( c=\sqrt{760.1584}\approx27.6 \)

Answer:

\( 27.6 \)