Question

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

Explanation:

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

In a triangle, the sum of angles is \( 180^{\circ} \). So, \( \angle R=180^{\circ}-130^{\circ}-30^{\circ}=20^{\circ} \).

Step2: Apply the Law of Sines

The Law of Sines states that \( \frac{\sin P}{q}=\frac{\sin Q}{r}=\frac{\sin R}{p} \). Here, \( P = 130^{\circ} \), \( Q = 30^{\circ} \), \( R = 20^{\circ} \), and \( p = 7 \). We use \( \frac{\sin Q}{p}=\frac{\sin P}{q} \).
Substitute the values: \( \frac{\sin30^{\circ}}{7}=\frac{\sin130^{\circ}}{q} \).
Since \( \sin30^{\circ}=\frac{1}{2}=0.5 \) and \( \sin130^{\circ}\approx0.766 \), we have \( \frac{0.5}{7}=\frac{0.766}{q} \).
Cross - multiply: \( 0.5q = 7\times0.766 \).

Step3: Solve for \( q \)

\( 0.5q=5.362 \), then \( q=\frac{5.362}{0.5}=10.724\approx10.7 \).

Answer:

\( q = 10.7 \)