Question

a triangular swimming pool measures 43 ft on one side and 32.8 ft on another side. the two sides form an angle that measures 41.3°. 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 to Use

We have two sides of a triangle and the included angle, so we use the Law of Cosines. The Law of Cosines formula for a triangle with sides \(a\), \(b\), and included angle \(C\) is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\). Let \(a = 43\) ft, \(b = 32.8\) ft, and \(C=41.3^{\circ}\).

Step2: Substitute the Values

Substitute the values into the formula:
\(c^{2}=43^{2}+32.8^{2}-2\times43\times32.8\times\cos(41.3^{\circ})\)
First, calculate \(43^{2}=1849\), \(32.8^{2}=1075.84\). Then, calculate \(2\times43\times32.8 = 2812.8\). Now, find \(\cos(41.3^{\circ})\approx0.752\) (using a calculator).
So, \(c^{2}=1849 + 1075.84-2812.8\times0.752\)

Step3: Calculate Each Part

Calculate \(2812.8\times0.752\approx2115.2256\)
Then, \(1849 + 1075.84=2924.84\)
Now, \(c^{2}=2924.84 - 2115.2256=809.6144\)

Step4: Find the Square Root

Take the square root of \(809.6144\) to find \(c\): \(c=\sqrt{809.6144}\approx28.5\)

Answer:

\(28.5\)