If a sixteen-foot ladder is leaning against a building, how far must its base be from the wall to achieve the correct slope?

To determine how far the base of a sixteen-foot ladder must be from the wall to achieve the correct slope, it is essential to apply the Pythagorean theorem. The ladder forms a right triangle with the wall and the ground, where the ladder itself is the hypotenuse.

When the ladder is in position, the height at which the ladder touches the wall, the distance from the base of the ladder to the wall, and the length of the ladder together form this right triangle. The Pythagorean theorem tells us that in a right triangle, the square of the hypotenuse (the length of the ladder) is equal to the sum of the squares of the other two sides (the height and the distance from the wall).

In this case, we set up the equation as follows:

1. Let the distance from the base of the ladder to the wall be represented as 'x'.
2. The height at which the ladder touches the wall will depend on that distance.
3. Given that the ladder length (the hypotenuse) is 16 feet, the relationship can be expressed as: ( x^2 + height^2 = 16^2 ).

For safety and optimal usage, a commonly recommended angle for