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

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). In (\triangle WXY), let (a = 10), (b = 12), (\theta=40^{\circ}), and (c = w). So (w^{2}=10^{2}+12^{2}-2\times10\times12\times\cos40^{\circ}).

Step2: Calculate the squared - terms and the cosine term

First, (10^{2}=100), (12^{2}=144), and (\cos40^{\circ}\approx0.766). Then (2\times10\times12\times\cos40^{\circ}=240\times0.766 = 183.84). And (100 + 144-183.84=100+144 - 183.84=60.16). So (w^{2}=60.16).

Step3: Find the value of (w)

Take the square - root of both sides: (w=\sqrt{60.16}\approx7.8).

Answer:

(7.8)