Name: Flosha, Ezckel G. 11 . Date: $$ 16 / 12 / 24 $$ Direction: Solve all problems neatly ( 40 pts ) A A ladder 12 cm long rests on a vertical wall. If the angle of elevation is $$ 30^{\circ} $$, find the distance the foot of the ladder is away from the wall. ( 4pts )

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**Solution:**

We are given a ladder that is 12 cm long resting against a vertical wall. The angle of elevation between the ladder and the ground is $$ 30^{\circ} $$. We need to find the horizontal distance ($$ x $$) from the foot of the ladder to the wall.

This forms a right-angled triangle where:
- The ladder is the hypotenuse ($$ h = 12 $$ cm).
- The angle of elevation is $$ \theta = 30^{\circ} $$.
- The horizontal distance is the adjacent side to the angle.

We can use the cosine of the angle to find the adjacent side:

$$
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}
$$

Plugging in the known values:

$$
\cos(30^{\circ}) = \frac{x}{12}
$$

We know that $$ \cos(30^{\circ}) = \frac{\sqrt{3}}{2} $$. Therefore:

$$
\frac{\sqrt{3}}{2} = \frac{x}{12}
$$

Solving for $$ x $$:

$$
x = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3} \text{ cm}
$$

**Answer:**
The foot of the ladder is $$ 6\sqrt{3} $$ centimeters away from the wall.
The foot of the ladder is $$ 6\sqrt{3} $$ cm away from the wall.

Name: Flosha, Ezckel G. 11 . Date: \( 16 / 12 / 24 \) Direction: Solve all problems neatly ( 40 pts ) A A ladder 12 cm long rests on a vertical wall. If the angle of elevation is \( 30^{\circ} \), find the distance the foot of the ladder is away from the wall. ( 4pts )

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