A 13-foot ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on level ground 12 feet from the base of the building. How high up the wall does the ladder reach? The ladder reaches up the wall. (Round to the nearest hundredth as needed.)

Step 1 ：Given a 13-foot ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on level ground 12 feet from the base of the building. We are asked to find how high up the wall does the ladder reach.

Step 2 ：This is a right triangle problem. We can use the Pythagorean theorem to solve it. 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 wall is one side (a), and the height up the wall is the other side (b). We are trying to find b.

Step 4 ：Given that \(a = 12\) and \(c = 13\), we can substitute these values into the Pythagorean theorem to solve for b.

Step 5 ：\(b = \sqrt{c² - a²} = \sqrt{13² - 12²} = 5.0\)

Step 6 ：Final Answer: The ladder reaches \(\boxed{5.0}\) feet up the wall.