From the top of a 60 m high cliff, the angle of depression to a boat in the sea is 30 degrees. How far is the boat from the base of the cliff?

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Q: From the top of a 60 m high cliff, the angle of depression to a boat in the sea is 30 degrees. How far is the boat from the base of the cliff?

Solution: Using tan(30°) = height/distance, we have distance = height/tan(30°) = 60/√3 = 60√3 m.

Steps: 10

Step 1: Understand the problem. We have a cliff that is 60 meters high and we need to find out how far the boat is from the base of the cliff.
Step 2: Identify the angle of depression. The angle of depression from the top of the cliff to the boat is 30 degrees.
Step 3: Visualize the situation. Imagine a right triangle where the height of the cliff is one side (60 m), the distance from the base of the cliff to the boat is the other side, and the line of sight to the boat 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 cliff) divided by the adjacent side (distance to the boat).
Step 5: Write the formula. For our triangle, tan(30°) = height / distance. We can rearrange this to find distance: distance = height / tan(30°).
Step 6: Substitute the values. We know the height is 60 m, so we plug that into the formula: distance = 60 / tan(30°).
Step 7: Calculate tan(30°). The value of tan(30°) is √3 / 3.
Step 8: Substitute tan(30°) into the formula. Now we have distance = 60 / (√3 / 3).
Step 9: Simplify the equation. This is the same as distance = 60 * (3 / √3).
Step 10: Calculate the distance. This simplifies to distance = 60√3 meters.