Questions: A 16-foot ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on level ground 5 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.)

Solution Steps

To find how high up the wall the ladder reaches, we can use the Pythagorean theorem. The ladder, the wall, and the ground form a right triangle. The ladder is the hypotenuse, the distance from the wall is one leg, and the height up the wall is the other leg. We can solve for the height using the formula \( a^2 + b^2 = c^2 \), where \( c \) is the length of the ladder, and \( a \) is the distance from the wall.

We have a right triangle formed by the ladder, the wall, and the ground. The ladder acts as the hypotenuse \( c \), the distance from the wall is one leg \( a \), and the height up the wall is the other leg \( b \).

Using the Pythagorean theorem, we can express the relationship between the sides of the triangle as follows: \[ a^2 + b^2 = c^2 \] Substituting the known values: \[ 5^2 + b^2 = 16^2 \]

Calculating the squares: \[ 25 + b^2 = 256 \] Rearranging gives: \[ b^2 = 256 - 25 = 231 \] Taking the square root: \[ b = \sqrt{231} \approx 15.1987 \] Rounding to the nearest hundredth, we find: \[ b \approx 15.20 \]

Final Answer