\begin{tabular}{l|l} 5. A b-foot ladder touches the side of a & 6 . \\ building at a point 5 feet above the ground. \\ At what height would a 15 -foot ladder touch \\ the building if it makes the same angle with & \\ the ground as the shorter ladder? \end{tabular}

To determine the height at which the 15-foot ladder will touch the building, we can use the concept of similar triangles since both ladders make the same angle with the ground. The height reached by the first ladder (5 feet) and its length (b feet) gives us a ratio that we can set equal to the ratio for the second ladder (15 feet). Using the similar triangles relationship: \[ \frac{h_1}{b} = \frac{h_2}{15} \] where \(h_1 = 5\) (the height reached by the first ladder) and \(h_2\) is the height reached by the second ladder. To calculate \(b\) (the length of the shorter ladder), we can think of this as a right triangle problem. The relationship can be further simplified to: \[ h_2 = \frac{h_1}{b} \times 15 \] If we assume \(b\) is a constant in relation to \(h_1\), let's say \(b = x\), then with \(h_1 = 5\), we can calculate the combined ratios and eventually find \(h_2\). Solving that gives you a similar height based on the properties of the triangles. If you need specific numbers, we can assign values and find exact ratios. In the previous relation, \[ \frac{5}{b} = \frac{h_2}{15} \] which simplifies our task. If we want numerical specifics, more information about \(b\) is needed to finalize the answer!