A 24 m long ladder is leaning against a wall. If the ladder makes an angle of $30^\circ$ with the wall, what is the distance (in m) of the foot of the ladder from the wall?

Question

Prepp Team · Accepted Answer

To solve this problem, we need to find the horizontal distance between the foot of the ladder and the wall.

We are given:

The length of the ladder (hypotenuse) is 24 meters.
The angle between the ladder and the wall is $30^\circ$.

We can model this problem as a right triangle where:

The ladder forms the hypotenuse.
The angle between the ladder and the wall is $30^\circ$.
The side opposite to this angle is the distance from the foot of the ladder to the wall (what we need to find).

From trigonometry, the cosine of an angle in a right triangle is defined as:

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

In this scenario, the adjacent side is the distance from the foot of the ladder to the wall, the hypotenuse is the length of the ladder, and $\theta = 30^\circ$.

Thus, we have:

$\cos(30^\circ) = \frac{x}{24}$

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

$\frac{\sqrt{3}}{2} = \frac{x}{24}$

To find $x$, we multiply both sides by 24:

$x = 24 \times \frac{\sqrt{3}}{2} = 12 \sqrt{3}$

However, note that we made a mistake earlier in defining the correct trigonometric function for this problem; we should actually use the sine function:

$\sin(30^\circ) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} = \frac{x}{24}$

Since $\sin(30^\circ) = \frac{1}{2}$, we have:

$\frac{1}{2} = \frac{x}{24}$

Solving for $x$:

$x = 24 \times \frac{1}{2} = 12$

Thus, the correct distance of the foot of the ladder from the wall is 12 meters, confirming the given correct answer.