A man is standing on a hill that is 80 m high. If he looks down at an angle of depression of 30 degrees, how far is he from the base of the hill?

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Q: A man is standing on a hill that is 80 m high. If he looks down at an angle of depression of 30 degrees, how far is he from the base of the hill?

Solution: Using tan(30°) = height/distance, we have 1/√3 = 80/distance. Therefore, distance = 80√3 ≈ 138.56 m.

Steps: 10

Step 1: Understand the problem. A man is on top of a hill that is 80 meters high and looks down at an angle of 30 degrees.
Step 2: Visualize the situation. Imagine a right triangle where the height of the hill is one side (80 m), the distance from the man to the base of the hill is the other side, and the line of sight from the man to the base of the hill is the hypotenuse.
Step 3: Identify the angle of depression. The angle of depression is the angle between the horizontal line from the man and the line of sight down to the base of the hill, which is 30 degrees.
Step 4: Use the tangent function. In a right triangle, the tangent of an angle is equal to the opposite side (height of the hill) divided by the adjacent side (distance from the base). So, tan(30°) = height/distance.
Step 5: Substitute the known values into the equation. We know the height is 80 m, so we write tan(30°) = 80/distance.
Step 6: Calculate tan(30°). The value of tan(30°) is 1/√3.
Step 7: Set up the equation: 1/√3 = 80/distance.
Step 8: Rearrange the equation to solve for distance: distance = 80 * √3.
Step 9: Calculate the distance. Using a calculator, 80 * √3 is approximately 138.56 m.
Step 10: Conclude that the man is approximately 138.56 meters from the base of the hill.