\begin{tabular}{l|l} A 11-foot ladder is leaning \\ against a building, with the base \\ of the ladder 3 feet from the \\ building. How high up on the \\ building will the top of the ladder \\ reach? & \( \begin{array}{l}\text { The ladder will reach } \\ \text { ft high up on the building. } \\ \text { (Simplify your answer. } \\ \text { Type an exact answer, } \\ \text { using radicals as } \\ \text { needed.) }\end{array} \)\end{tabular}

To solve the problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the ladder in this case) is equal to the sum of the squares of the other two sides. Here, the ladder is the hypotenuse with a length of 11 feet, the distance from the base of the ladder to the building is one side measuring 3 feet, and we want to find the height (h) that the ladder reaches on the building, which is the other side of the triangle. According to the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] Where: - \( a = 3 \) (the distance from the wall) - \( b = h \) (the height the ladder reaches) - \( c = 11 \) (the length of the ladder) Plugging in the values: \[ 3^2 + h^2 = 11^2 \] This simplifies to: \[ 9 + h^2 = 121 \] Now, isolate \( h^2 \): \[ h^2 = 121 - 9 \] \[ h^2 = 112 \] Taking the square root of both sides gives: \[ h = \sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7} \] So, the top of the ladder will reach **4√7 feet high up on the building.**