The base of a 13 -foot ladder is 4 feet from a building. If the ladder reaches the flat roof, how tall is the building?

Step 1 ：We are given a right triangle problem. The base of a 13-foot ladder is 4 feet from a building. If the ladder reaches the flat roof, we are asked to find the height of the building.

Step 2 ：We can use the Pythagorean theorem to solve this problem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as: \(a² + b² = c²\).

Step 3 ：In this case, the length of the ladder is the hypotenuse (c), the distance from the base of the ladder to the building is one side (a), and the height of the building is the other side (b). We are trying to find b.

Step 4 ：Given that \(a = 4\) and \(c = 13\), we can substitute these values into the Pythagorean theorem to solve for b.

Step 5 ：By calculating, we find that \(b = \sqrt{c² - a²} = \sqrt{13² - 4²} = \sqrt{169 - 16} = \sqrt{153} = 12.37\).

Step 6 ：Final Answer: The height of the building is approximately \(\boxed{12.37}\) feet.