Final answer:
The minimum distance from the wall for a 15-meter ladder to be safe is found by calculating X = 15 * cos(π/12), using trigonometry to determine the horizontal distance required.
Step-by-step explanation:
To determine the minimum distance from the wall that the foot of a 15-meter ladder can be placed while maintaining safety, which is defined by forming an angle greater than π/12 radians with the wall, we use trigonometric functions derived from the right-angled triangle formed by the ladder, the wall, and the ground.
Firstly, since the ladder makes an angle with the wall, we look at the cosine of that angle to find the horizontal distance from the wall (adjacent side of the triangle).
The compound angle formula is not directly needed here, but for the sake of using trigonometric identities, let's consider the cosine function. The cosine of π/12 represents the ratio of the adjacent side over the hypotenuse (the ladder's length). To find the minimum safe horizontal distance, X, from the wall, we calculate:
X = L * cos(π/12)
where L is the length of the ladder, 15 meters, and π/12 is the minimum safe angle.
Using this equation, we find that:
X = 15 * cos(π/12)
To obtain an exact expression, we must calculate cos(π/12). Since exact values of cos(π/12) are not typically known, this could be left in terms of the cosine function, which in this case is a safe distance calculation using trigonometry.