A 14-foot ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on level ground 10 feet from the base of the building.

How high up the wall does the ladder reach?

Step 1 ：A 14-foot ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on level ground 10 feet from the base of the building. We are asked to find how high up the wall does the ladder reach.

Step 2 ：We can solve this problem using the Pythagorean theorem, which states that in a right-angled 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 \(c^2 = a^2 + b^2\), where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

Step 3 ：In this case, the length of the ladder represents the hypotenuse (c), the distance from the base of the building to the ladder represents one side of the triangle (a), and the height up the wall that the ladder reaches represents the other side of the triangle (b). We know that \(a = 10\) feet and \(c = 14\) feet, and we are asked to find b.

Step 4 ：We can rearrange the Pythagorean theorem to solve for b: \(b = \sqrt{c^2 - a^2}\).

Step 5 ：Substituting the known values into the formula, we get \(b = \sqrt{14^2 - 10^2}\).

Step 6 ：Calculating the above expression, we find that \(b \approx 9.8\) feet.

Step 7 ：So, the ladder reaches approximately \(\boxed{9.8}\) feet up the wall.