Question

find z.
image of a right triangle with a 30° angle, a 60° angle, one leg labeled z, the other leg labeled 3√3 mi, and a right angle
write your answer in simplest radical form.
blank miles

Explanation:

Step1: Identify trigonometric ratio

We use tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. For the $30^\circ$ angle, opposite side is $z$, adjacent is $3\sqrt{3}$.

Step2: Substitute values into formula

$\tan(30^\circ) = \frac{z}{3\sqrt{3}}$
We know $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, so:
$\frac{1}{\sqrt{3}} = \frac{z}{3\sqrt{3}}$

Step3: Solve for z

Multiply both sides by $3\sqrt{3}$:
$z = \frac{1}{\sqrt{3}} \times 3\sqrt{3}$
Simplify: $z = 3$

Answer:

3 miles