What is the maximum height that a 24 foot ladder can safely reach if the base of the ladder should always be about 1 foot away from the vertical support for every 4 feet of height?

The maximum height it can safely reach is 23 feet (approximately)

Step-by-step explanation: First of all the ladder is 24 feet long. Then it needs to rest on a vertical support and it is recommended that for every 4 feet the ladder goes up, there should be a 1 foot safety distance between the base of the ladder and the vertical support. In other words, if the whole length of the ladder were to be placed on the vertical support, the 24-foot height must have a safety distance of,

Base = height/4

Base = 24/4

Base = 6

What this implies is that, as long as the whole length of the ladder (24 ft) is placed on the support, the safe distance from the base of the support would be 6 feet. At this rate, the maximum height that the ladder can reach is simply the vertical support itself. This results in a right angled triangle with hypotenuse 24 ft, and one side measuring 6ft. The unknown side can be derived by use of the Pythagoras theorem which states that,

AC^2 = AB^2 + BC^2

Where AC is the hypotenuse and AB and BC are the other two sides.

We can now substitute for the values as follows,

24^2 = 6^2 + BC^2

576 = 36 + BC^2

Subtract 36 from both sides of the equation

540 = BC^2

Add the square root sign to both sides of the equation

BC = 23.2379

By approximation, BC equals 23 feet.

That means the maximum height the ladder can SAFELY reach is 23 feet.