Question

in \\( \triangle nop \\), \\( p = 670 \\) inches, \\( n = 330 \\) inches and \\( \angle o = 47 ^ { \circ } \\). find the length of \\( o \\), to the nearest inch.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(o^{2}=n^{2}+p^{2}-2np\cos O\).
Substitute \(n = 330\), \(p = 670\), and \(O = 47^{\circ}\) into the formula:
\(o^{2}=330^{2}+670^{2}-2\times330\times670\times\cos(47^{\circ})\).
First, calculate \(330^{2}=108900\), \(670^{2}=448900\), and \(2\times330\times670 = 442200\).
\(\cos(47^{\circ})\approx0.682\).
Then \(o^{2}=108900 + 448900-442200\times0.682\).

Step2: Calculate the value of \(o^{2}\)

\(108900+448900 = 557800\).
\(442200\times0.682=442200\times(0.6 + 0.08+0.002)=442200\times0.6+442200\times0.08 + 442200\times0.002=265320+35376+884.4 = 301580.4\).
\(o^{2}=557800-301580.4 = 256219.6\).

Step3: Find the value of \(o\)

Take the square - root of \(o^{2}\): \(o=\sqrt{256219.6}\approx506.2\).

Answer:

\(506\) inches