From the top of a 100-meter high cliff, the angle of depression to a boat in the
Practice Questions
Q1
From the top of a 100-meter 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?
100 meters
150 meters
173 meters
200 meters
Questions & Step-by-Step Solutions
From the top of a 100-meter 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?
Correct Answer: 173.21 meters
Step 1: Understand the problem. We have a cliff that is 100 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 (100 meters), the distance from the base of the cliff to the boat is the other side, and the line of sight from the top of the cliff 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 divided by the adjacent side. Here, the opposite side is the height of the cliff (100 meters) and the adjacent side is the distance we want to find.
Step 5: Write the formula. The formula is: tan(angle) = opposite / adjacent. We can rearrange it to find the adjacent side: adjacent = opposite / tan(angle).
Step 6: Plug in the values. We have the opposite side as 100 meters and the angle as 30 degrees. So, we need to calculate: adjacent = 100 / tan(30 degrees).
Step 7: Calculate tan(30 degrees). The value of tan(30 degrees) is 1/√3.
Step 8: Substitute tan(30 degrees) into the formula. Now we have: adjacent = 100 / (1/√3).
Step 9: Simplify the equation. Dividing by a fraction is the same as multiplying by its reciprocal: adjacent = 100 * √3.
Step 10: Calculate the final distance. Using the approximate value of √3 (about 1.732), we find that the distance is approximately 173.21 meters.
Trigonometry – The problem involves using the tangent function to relate the height of the cliff to the distance from the base of the cliff to the boat.
Angle of Depression – Understanding the angle of depression and how it relates to the horizontal distance and vertical height.
