Question

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

Explanation:

Step1: Find the measure of angle \( X \)

Use the triangle - angle sum theorem (\( \angle W+\angle Y+\angle X = 180^{\circ} \)).
Given \( \angle W = 83^{\circ} \) and \( \angle Y=30^{\circ} \), then \( \angle X=180^{\circ}-(83^{\circ}+30^{\circ})=67^{\circ} \).

Step2: Apply the Law of Sines

The Law of Sines states that \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \). In \( \triangle WXY \), if \( a = w \), \( A=\angle W = 83^{\circ} \), \( b = 9 \), and \( B=\angle Y = 30^{\circ} \), then \( \frac{w}{\sin83^{\circ}}=\frac{9}{\sin30^{\circ}} \).
Since \( \sin30^{\circ}=\frac{1}{2} = 0.5 \) and \( \sin83^{\circ}\approx0.9925 \), we can rewrite the equation as \( w=\frac{9\times\sin83^{\circ}}{\sin30^{\circ}} \).
Substitute the values: \( w=\frac{9\times0.9925}{0.5} \).
First, calculate \( 9\times0.9925 = 8.9325 \).
Then, \( w=\frac{8.9325}{0.5}=17.865\approx17.9 \).

Answer:

\( 17.9 \)