Question

solving real - world problems with 30° - 60° - 90° triangles
from the side view, a gymnastics mat forms a right triangle with other angles measuring 60° and 30°. the gymnastics mat extends 5 feet across the floor. how high is the mat off the ground?
options:

• $\frac{5}{2}$ ft
• $\frac{5sqrt{3}}{3}$ ft
• $5sqrt{3}$
• 10

(there is a right triangle diagram with one leg labeled height, the other leg labeled 5 ft, one acute angle 30° and the other 60°)

Explanation:

Step1: Identify trigonometric ratio

We use tangent: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$
Here, $\theta=30^\circ$, opposite side = height, adjacent side = 5 ft.

Step2: Substitute values into formula

$\tan(30^\circ)=\frac{\text{height}}{5}$
We know $\tan(30^\circ)=\frac{\sqrt{3}}{3}$, so:
$\frac{\sqrt{3}}{3}=\frac{\text{height}}{5}$

Step3: Solve for height

Multiply both sides by 5:
$\text{height}=5\times\frac{\sqrt{3}}{3}=\frac{5\sqrt{3}}{3}$

Answer:

$\frac{5\sqrt{3}}{3}$ ft