From the top of a tower, the angle of depression to a point on the ground is 30 degrees. If the height of the tower is 50 meters, how far is the point from the base of the tower?
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From the top of a tower, the angle of depression to a point on the ground is 30 degrees. If the height of the tower is 50 meters, how far is the point from the base of the tower?
Q: From the top of a tower, the angle of depression to a point on the ground is 30 degrees. If the height of the tower is 50 meters, how far is the point from the base of the tower?
Step 1: Understand the problem. We have a tower that is 50 meters tall, and we need to find out how far a point on the ground is from the base of the tower.
Step 2: Identify the angle of depression. The angle of depression from the top of the tower to the point on the ground is 30 degrees.
Step 3: Visualize the situation. Imagine a right triangle where the height of the tower is one side (50 meters), the distance from the base of the tower to the point on the ground is the other side, and the line of sight from the top of the tower 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 tower) divided by the adjacent side (distance from the base). So, tan(30 degrees) = height / distance.
Step 5: Rearrange the formula to find the distance. Distance = height / tan(30 degrees).
Step 6: Calculate tan(30 degrees). The value of tan(30 degrees) is 1/√3.
Step 7: Substitute the values into the formula. Distance = 50 / (1/√3).
Step 8: Simplify the calculation. Dividing by a fraction is the same as multiplying by its reciprocal, so Distance = 50 * √3.
Step 9: Final answer. The distance from the base of the tower to the point on the ground is 50√3 meters.
