Question

a 10.6-foot ladder is leaning against a wall. the top of the ladder forms a $30^{circ}$ angle with the wall. how far is the bottom of the ladder from the wall?

Explanation:

Step1: Identify right triangle components

The ladder is the hypotenuse ($c=10.6$ ft), the distance from the bottom of the ladder to the wall is the side opposite the $30^\circ$ angle ($a$), and the wall forms the adjacent side.

Step2: Apply sine function

Sine of an angle in a right triangle is $\frac{\text{opposite}}{\text{hypotenuse}}$.
$\sin(30^\circ) = \frac{a}{10.6}$

Step3: Solve for $a$

Substitute $\sin(30^\circ)=0.5$ and rearrange.
$a = 10.6 \times \sin(30^\circ) = 10.6 \times 0.5$

Answer:

5.3 feet