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 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: Apply the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Let \(a = 46\), \(b=32.6\), and \(C = 41.1^{\circ}\).
Substitute the values into the formula:
\(c^{2}=46^{2}+32.6^{2}-2\times46\times32.6\times\cos(41.1^{\circ})\)
First, calculate \(46^{2}=2116\), \(32.6^{2}=1062.76\).
Then, calculate \(2\times46\times32.6 = 2999.2\).
\(\cos(41.1^{\circ})\approx0.7547\)
So, \(2999.2\times0.7547\approx2265.4\)
Now, \(c^{2}=2116 + 1062.76-2265.4\)
\(c^{2}=3178.76 - 2265.4\)
\(c^{2}=913.36\)

Step2: Solve for \(c\)

Take the square root of both sides: \(c=\sqrt{913.36}\)
\(c\approx30.2\)

Answer:

\(30.2\)