A ladder 10 meters long leans against a wall, forming a 60-degree angle with the ground. How high up the wall does it reach?
["How High Does a 10-Meter Ladder Reach When Leaning at a 60-Degree Angle? A Clear Mathematical Guide", "When a 10-meter ladder leans against a wall at a 60-degree angle with the ground, many people wonder exactly how high it reaches up the vertical surface. Understanding this scenario involves basic trigonometry — specifically, using the sine function to calculate vertical height based on the angle and length of the ladder.", "### The Physics and Math Behind the Problem", "The ladder acts as the hypotenuse of a right triangle, with the wall forming the vertical side and the ground the horizontal base. The angle between the ladder and the ground is 60 degrees. To find the vertical height ( h ) the ladder reaches on the wall, we use:", "[\n\sin(\ heta) = \frac{\ ext{opposite}}{\ ext{hypotenuse}} = \frac{h}{10}\n]", "Where:", "- ( \ heta = 60^\circ )\n- Hypotenuse (ladder length) = 10 meters\n- Opposite side (height up the wall) = ( h )", "### Step-by-Step Calculation", "1. Use the sine function:\n [\n \sin(60^\circ) = \frac{h}{10}\n ]", "2. We know from trigonometric values that:\n [\n \sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866\n ]", "3. Substitute into the equation:\n [\n \frac{\sqrt{3}}{2} = \frac{h}{10}\n ]", "4. Solve for ( h ):\n [\n h = 10 \ imes \frac{\sqrt{3}}{2} = 5\sqrt{3}\n ]", "5. Calculate the numerical value:\n [\n h \approx 5 \ imes 1.732 = 8.66 \ ext{ meters}\n ]", "### Final Result", "The top of a 10-meter ladder leaning at a 60-degree angle reaches approximately 8.66 meters up the wall. This precise value comes from ( 5\sqrt{3} ) meters, a clean exact form, or about 8.66 meters when rounded.", "### Why This Knowledge Matters", "Knowing how angles and lengths relate in real-world setups — such as ladder placement, construction, or rescue operations — ensures safety and efficiency. It helps avoid overreaching, unstable climbing, or underestimating heights, reducing risks in both home and commercial environments.", "In summary:\nWith a 10-meter ladder at 60°, the height along the wall is Exactly ( 5\sqrt{3} ) meters, or about 8.66 meters — perfect for planning safe and accurate ladder use.", "---", "Keywords: ladder height calculation, 10-meter ladder at 60 degrees, how high does ladder reach, trigonometry ladder height, trig on walls, real-world ladder math"]
