Question

a triangular swimming pool measures 43 ft on one side and 32.2 ft on another side. the two sides form an angle that measures 40.5°. 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

For a triangle with sides \(a\), \(b\), and included angle \(C\), the Law of Cosines is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\). Let \(a = 43\), \(b = 32.2\), and \(C=40.5^{\circ}\).

Step2: Substitute the values into the formula

First, calculate \(a^{2}=43^{2}=1849\), \(b^{2}=32.2^{2}=1036.84\), and \(2ab\cos(C)=2\times43\times32.2\times\cos(40.5^{\circ})\). Calculate \(\cos(40.5^{\circ})\approx0.758\). Then \(2\times43\times32.2\times0.758 = 2\times43\times32.2\times0.758\approx2\times43\times24.4076\approx2\times1049.5268\approx2099.0536\).
Now, \(c^{2}=1849 + 1036.84-2099.0536=1849+1036.84 = 2885.84; 2885.84 - 2099.0536 = 786.7864\).

Step3: Find \(c\)

Take the square root of \(c^{2}\): \(c=\sqrt{786.7864}\approx28.05\approx28.1\) (rounded to the nearest tenth).

Answer:

28.1