\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 solve this problem, we can use trigonometry to find the height at which a 15-foot ladder would touch the building if it makes the same angle with the ground as the shorter ladder. Let's denote: - \( b \) as the length of the shorter ladder, - \( h \) as the height at which the 15-foot ladder touches the building. Given: - The shorter ladder touches the building at a point 5 feet above the ground. We can use the tangent function to relate the angle, the height of the shorter ladder, and the length of the shorter ladder. The tangent of an angle is equal to the ratio of the opposite side (height) to the adjacent side (length). For the shorter ladder: \[ \tan(\theta) = \frac{5}{b} \] For the 15-foot ladder: \[ \tan(\theta) = \frac{h}{15} \] Since the angle is the same for both ladders, we can set the two expressions equal to each other: \[ \frac{5}{b} = \frac{h}{15} \] Now, we can solve for \( h \) to find the height at which the 15-foot ladder would touch the building. Solve the equation by following steps: - step0: Solve for \(h\): \(\frac{5}{b}=\frac{h}{15}\) - step1: Swap the sides: \(\frac{h}{15}=\frac{5}{b}\) - step2: Multiply both sides of the equation by \(15:\) \(\frac{h}{15}\times 15=\frac{5}{b}\times 15\) - step3: Multiply the terms: \(h=\frac{5\times 15}{b}\) - step4: Evaluate: \(h=\frac{75}{b}\) The height at which a 15-foot ladder would touch the building if it makes the same angle with the ground as the shorter ladder is given by the expression \( h = \frac{75}{b} \), where \( b \) is the length of the shorter ladder.