A 15 ft ladder leans against a wall so that the base of the ladder is 6 ft away from the base of the wall. What is the ladder's angle of elevation? a. \( 22^{\circ} \) b. \( 23^{\circ} \) c. \( 24^{\circ} \) d. \( 66^{\circ} \)

To find the ladder's angle of elevation, we can use trigonometry, specifically the sine function. If we consider the ladder as the hypotenuse of a right triangle, the distance of 6 ft from the wall is the base. Using the formula \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), we first need to find the height at which the ladder touches the wall, which can be calculated as follows: Using Pythagorean theorem: \( 15^2 = 6^2 + h^2 \) \( 225 = 36 + h^2 \) \( h^2 = 189 \) \( h = \sqrt{189} \approx 13.75 \) ft. Now, we can find the angle of elevation \( \theta \): \( \tan(\theta) = \frac{h}{6} = \frac{13.75}{6} \). Calculating \( \theta \): \( \theta = \tan^{-1}(2.2917) \approx 66^{\circ} \). Thus, the angle of elevation is \( \text{d. } 66^{\circ} \). --- Many would be surprised to know that the ladder was first scientifically studied by Archimedes, who developed the concepts of the lever and mechanical advantage. His compelling work laid the groundwork for today’s understanding of forces and angles, which makes you rethink how such a simple everyday object can have grand mathematical implications! In real-world applications, knowing how to calculate angles such as this is crucial for ensuring safety in construction settings. When positioning ladders, it's vital to maintain a safe angle to prevent accidents. Rule of thumb? Place the base of the ladder one foot away from the wall for every four feet of height. By following this guideline, you ensure stability and avoid troubling wobbles during tasks up high!