Question

find z.
write your answer in simplest radical form.
miles

Explanation:

Step1: Identify trigonometric ratio

We use the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. For the $30^\circ$ angle, $z$ is opposite, and $14\sqrt{3}$ is adjacent.
$\tan(30^\circ) = \frac{z}{14\sqrt{3}}$

Step2: Substitute $\tan(30^\circ)$ value

We know $\tan(30^\circ) = \frac{1}{\sqrt{3}}$. Substitute this into the equation:
$\frac{1}{\sqrt{3}} = \frac{z}{14\sqrt{3}}$

Step3: Solve for $z$

Multiply both sides by $14\sqrt{3}$:
$z = \frac{1}{\sqrt{3}} \times 14\sqrt{3}$
Simplify: $z = 14$

Answer:

14 miles