Question

a triangular swimming pool measures 46 ft on one side and 32.6 ft on another side. the two sides form an angle that measures 40.6°. 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 of a triangle, \( a = 46 \) ft, \( b = 32.6 \) ft, and the included angle \( C = 40.6^\circ \). The Law of Cosines for the third side \( c \) is \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \).

Step2: Substitute the values

First, calculate \( a^{2}=46^{2} = 2116 \), \( b^{2}=32.6^{2}=1062.76 \), and \( 2ab = 2\times46\times32.6=3027.2 \). Then, \( \cos(40.6^\circ)\approx\cos(40.6)\approx0.757 \). So, \( 2ab\cos(C)=3027.2\times0.757\approx2291.6 \).
Now, \( c^{2}=2116 + 1062.76- 2291.6=2116 + 1062.76=3178.76; 3178.76 - 2291.6 = 887.16 \).

Step3: Take the square root

\( c=\sqrt{887.16}\approx29.8 \) (rounded to the nearest tenth).

Answer:

\( 29.8 \)