A ladder leans against a wall, forming a 75-degree angle with the ground. If the ladder is 20 feet long, how high up the wall does it reach?
["How to Calculate the Height a 20-Foot Ladder Reaches on a Wall at a 75-Degree Angle", "When a ladder leans against a wall, forming a precise angle with the ground, understanding how high it reaches is important for safety, proper placement, and avoiding damage. Whether you're a homeowner, a DIY enthusiast, or a contractor, knowing how to calculate the vertical reach can help you plan efficiently.", "### The Math Behind the Ladder’s Angle and Height", "In this common real-world scenario, the ladder forms a right triangle with the wall and the ground. The ladder acts as the hypotenuse (20 feet), and the vertical height it reaches on the wall is the opposite side of the angle between the ladder and the ground.", "The key trigonometric function here is the sine, defined as:", "> sin(θ) = opposite / hypotenuse", "Where:\n- θ is the angle between the ladder and the ground (75°)\n- opposite is the height on the wall\n- hypotenuse is the ladder length (20 feet)", "### Step-by-Step Calculation", "1. Set up the sine formula:\n [\n \sin(75^\circ) = \frac{\ ext{height}}{\ ext{length}} = \frac{h}{20}\n ]", "2. Solve for height (h):\n [\n h = 20 \ imes \sin(75^\circ)\n ]", "3. Use a calculator for accuracy:\n The sine of 75 degrees is approximately 0.9659.\n [\n h = 20 \ imes 0.9659 = 19.318 \ ext{ feet}\n ]", "### Conclusion", "At a 75-degree angle, a 20-foot ladder reaches approximately 19.32 feet up the wall. This near-vertical position means careful placement is crucial—any slip slightly higher could mean serious risk.", "### Why This Matters", "- Safety: Avoid overreaching; falling from higher offsets increases injury risk.\n- Placement: Ensures the ladder stays 1 foot away from the wall for every foot climbed (the classic “4:1 rule”).\n- Efficiency: Knowing exact reach helps with ladder sizing and positioning long before climbing.", "In summary, using trigonometry, a 20-foot ladder makes contact with the wall around 19.3 feet high when leaning at a 75-degree angle. This simple calculation combines physics, math, and practical knowledge—key for safe and effective use of ladders in daily life."]
