Question

x = type your answer_

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite to side \(c\) is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). Here, let \(a = 8\), \(b = 9\), \(\theta=47^{\circ}\), and \(c=x\). So \(x^{2}=8^{2}+9^{2}-2\times8\times9\times\cos(47^{\circ})\).

Step2: Calculate the values

First, \(8^{2}=64\), \(9^{2}=81\). And \(\cos(47^{\circ})\approx0.682\). Then \(2\times8\times9\times\cos(47^{\circ})=144\times0.682 = 98.208\). So \(x^{2}=64 + 81-98.208\).

Step3: Simplify the right - hand side

\(x^{2}=145 - 98.208=46.792\).

Step4: Solve for \(x\)

Take the square - root of both sides: \(x=\sqrt{46.792}\approx6.84\).

Answer:

\(x\approx6.84\)