Question

what is the length of side q, given r = 20, s = 30, and q = 15°? round the answer to the nearest tenth.
not drawn to scale
○ 11.9
○ 40.7

Explanation:

Step1: Identify the Law of Cosines formula

For a triangle with sides \( r \), \( s \), and \( q \), and the angle \( Q \) between sides \( r \) and \( s \), the Law of Cosines states that \( q^2 = r^2 + s^2 - 2rs\cos(Q) \).

Step2: Substitute the given values

We know that \( r = 20 \), \( s = 30 \), and \( Q = 15^\circ \). Substituting these values into the formula:
\[
q^2 = 20^2 + 30^2 - 2 \times 20 \times 30 \times \cos(15^\circ)
\]

Step3: Calculate each term

First, calculate \( 20^2 = 400 \), \( 30^2 = 900 \). Then, calculate \( 2 \times 20 \times 30 = 1200 \). The value of \( \cos(15^\circ) \approx 0.9659 \).
\[
q^2 = 400 + 900 - 1200 \times 0.9659
\]
\[
q^2 = 1300 - 1159.08
\]
\[
q^2 = 140.92
\]

Step4: Solve for \( q \)

Take the square root of \( q^2 \):
\[
q = \sqrt{140.92} \approx 11.9
\]

Answer:

11.9