Solving for a Side Length in a Right Triangle Using Law of Sines
To solve for \( w \), we'll use the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides.
Given triangle has angles 30°, 60°, and because the sum of angles in a triangle is always 180°, the remaining angle is 90°, making it a right triangle. The side opposite the 30° angle is half the hypotenuse in a right triangle, while the side opposite the 60° angle is equal to \( \sqrt{3}/2 \) times the hypotenuse.
Since we have:
\[
\frac{w}{\sin(90°)} = \frac{10}{\sin(60°)}
\]
We know that \( \sin(90°) = 1 \) and \( \sin(60°) = \sqrt{3}/2 \), so we have:
\[
w = 10 \times \frac{1}{\sqrt{3}/2} = 10 \times \frac{2}{\sqrt{3}} = \frac{20}{\sqrt{3}}
\]
To express in simplest radical form, we rationalize the denominator:
\[
w = \frac{20}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{20\sqrt{3}}{3}
\]
Thus, \( w = \frac{20\sqrt{3}}{3} \) meters in simplest radical form.
