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?
The height of the building is $\mathrm{ft}$. (Simplify your answer. Type an exact answer, using radicals as needed.)
The height of the building is approximately $\mathrm{ft}$. (Round to the nearest tenth as needed.)

Step 1 ：This problem involves a right triangle, where 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 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 ：Given that a = 4 and c = 13, we can substitute these values into the equation to find b.

Step 4 ：Solving for b, we get \(b = \sqrt{c² - a²} = \sqrt{13² - 4²} = \sqrt{169 - 16} = \sqrt{153}\).

Step 5 ：The exact height of the building is \(\sqrt{153}\) feet.

Step 6 ：To find the approximate height, we can round \(\sqrt{153}\) to the nearest tenth, which gives us approximately 12.37 feet.

Step 7 ：Final Answer: The height of the building is approximately \(\boxed{12.37}\) feet.