A person is standing on a hill 100 meters high. If he looks at a point on the ground at an angle of depression of 30 degrees, how far is the point from the base of the hill?
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A person is standing on a hill 100 meters high. If he looks at a point on the ground at an angle of depression of 30 degrees, how far is the point from the base of the hill?
Q: A person is standing on a hill 100 meters high. If he looks at a point on the ground at an angle of depression of 30 degrees, how far is the point from the base of the hill?
Step 1: Understand the problem. A person is on a hill that is 100 meters high and looks down at a point on the ground.
Step 2: Identify the angle of depression. The angle of depression is given as 30 degrees.
Step 3: Visualize the situation. Imagine a right triangle where the height of the hill is one side (100 meters), the distance from the base of the hill to the point on the ground is the other side, and the line of sight from the person to the point on the ground is the hypotenuse.
Step 4: Use the tangent function. In a right triangle, the tangent of an angle is the opposite side (height of the hill) divided by the adjacent side (distance from the base).
Step 5: Set up the equation. We have tan(30°) = height / distance. We can rearrange this to find distance: distance = height / tan(30°).
Step 6: Plug in the values. The height is 100 meters, so distance = 100 / tan(30°).
Step 7: Calculate tan(30°). The value of tan(30°) is √3 / 3.
Step 8: Substitute tan(30°) into the equation. So, distance = 100 / (√3 / 3).
Step 9: Simplify the equation. This is the same as distance = 100 * (3 / √3) = 100√3 meters.
Step 10: Conclusion. The distance from the base of the hill to the point on the ground is 100√3 meters.
