Question

find s.

write your answer as an integer or as a decimal rounded to the nearest tenth.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \( s^{2}=a^{2}+b^{2}-2ab\cos C\). Here, \(a = 5\), \(b = 10\), and \(C=135^{\circ}\), \(\cos(135^{\circ})=-\frac{\sqrt{2}}{2}\approx - 0.707\).
Substitute the values into the formula: \(s^{2}=5^{2}+10^{2}-2\times5\times10\times\cos(135^{\circ})\).

Step2: Calculate each term

First, \(5^{2}=25\), \(10^{2}=100\), and \(2\times5\times10 = 100\).
So \(s^{2}=25 + 100-100\times(-0.707)\).
\(s^{2}=25+100 + 70.7\).

Step3: Find \(s^{2}\) and then \(s\)

\(s^{2}=195.7\).
Take the square - root of both sides: \(s=\sqrt{195.7}\approx14.0\).

Answer:

\(s = 14.0\)