From the top of a 40-meter high tower, the angle of depression to a point on the
Practice Questions
Q1
From the top of a 40-meter high tower, the angle of depression to a point on the ground is 60 degrees. How far is the point from the base of the tower?
20√3 meters
40 meters
30 meters
50 meters
Questions & Step-by-Step Solutions
From the top of a 40-meter high tower, the angle of depression to a point on the ground is 60 degrees. How far is the point from the base of the tower?
Correct Answer: 40√3 meters
Step 1: Understand the problem. We have a tower that is 40 meters high.
Step 2: The angle of depression from the top of the tower to a point on the ground is 60 degrees.
Step 3: Visualize the situation. Draw a right triangle where the tower is the vertical side (height) and the distance from the base of the tower to the point on the ground is the horizontal side (base).
Step 4: The angle of depression is the angle between the horizontal line from the top of the tower and the line of sight to the point on the ground.
Step 5: In this triangle, the height of the tower (40 meters) is the opposite side to the angle of depression (60 degrees).
Step 6: The distance from the base of the tower to the point on the ground is the adjacent side.
Step 7: Use the tangent function, which relates the opposite side to the adjacent side: tan(angle) = opposite / adjacent.
Step 8: Rearrange the formula to find the distance (adjacent side): distance = height / tan(angle).
Step 9: Substitute the values into the formula: distance = 40 / tan(60 degrees).
Step 10: Calculate tan(60 degrees), which is √3. So, distance = 40 / (√3).
Step 11: To simplify, multiply by the reciprocal: distance = 40 * (1/√3) = 40√3 meters.
Trigonometry – The problem involves using the tangent function to relate the height of the tower and the angle of depression to find the horizontal distance.
Angle of Depression – Understanding that the angle of depression from the top of the tower to the point on the ground is equal to the angle of elevation from the point on the ground to the top of the tower.
Right Triangle Properties – The scenario can be visualized as a right triangle where the height of the tower is one leg, the distance from the base of the tower to the point is the other leg, and the line of sight is the hypotenuse.
