# A 6-foot ladder touches the side of a building at a point 5 feet above the ground. At what height would a 15-foot ladder touch the building if it makes the same angle with the ground as the shorter ladder?

Feb 3, 2016

The heigt is $12.5$ foot.

#### Explanation:

Here's a simple sketch of your problem:

In both cases, your angle $\alpha$ is the same.

The ladder, the ground and the building build a right-angle triangle where the respective ladder is the hypotenuse.

We know that $\sin$ of an angle can be computed as the opposite side divided by the hypotenuse.

Thus, in the case of the $6$-foot ladder, we have:

$\sin \alpha = \frac{5}{6}$

For the $15$-foot ladder, we have

$\sin \alpha = \frac{x}{15}$

However, the angles are the same. Thus, we get

$\frac{5}{6} = \frac{x}{15} \text{ "<=>" " x = 5/6 * 15 = 75/6 = 12.5 " foot}$