Question

a triangular swimming pool measures 43 ft on one side and 32.1 ft on another side. the two sides form an angle that measures 41.1°. 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

We have two sides \( a = 43 \) ft, \( b = 32.1 \) ft and the included angle \( C = 41.1^\circ \). The Law of Cosines for the third side \( c \) is \( c^2=a^2 + b^2-2ab\cos(C) \).

Step2: Substitute the values

Substitute \( a = 43 \), \( b = 32.1 \), and \( C = 41.1^\circ \) into the formula. First, calculate \( a^2=43^2 = 1849 \), \( b^2 = 32.1^2=1030.41 \), and \( 2ab\cos(C)=2\times43\times32.1\times\cos(41.1^\circ) \). Calculate \( \cos(41.1^\circ)\approx0.753 \), then \( 2\times43\times32.1\times0.753\approx2\times43\times32.1\times0.753 = 86\times32.1\times0.753\approx86\times24.1713\approx2078.7318 \).

Step3: Calculate \( c^2 \)

\( c^2=1849 + 1030.41-2078.7318=1849 + 1030.41=2879.41;2879.41 - 2078.7318 = 800.6782 \)

Step4: Find \( c \)

Take the square root of \( c^2 \): \( c=\sqrt{800.6782}\approx28.3 \)

Answer:

\( 28.3 \)