A person is standing on a hill 80 m high. The angle of depression to a car on th

A person is standing on a hill 80 m high. The angle of depression to a car on the ground is 60 degrees. How far is the car from the base of the hill?

A person is standing on a hill 80 m high. The angle of depression to a car on the ground is 60 degrees. How far is the car from the base of the hill?

Step 1: Understand the problem. A person is on a hill that is 80 meters high and sees a car on the ground at an angle of 60 degrees downwards.
Step 2: Visualize the situation. Draw a right triangle where the height of the hill is one side (80 m), the distance from the base of the hill to the car is the other side, and the line of sight to the car is the hypotenuse.
Step 3: Identify the relevant angle. The angle of depression from the person to the car is 60 degrees.
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 to the car).
Step 5: Write the equation using the tangent function: tan(60°) = height / distance.
Step 6: Substitute the known values into the equation: tan(60°) = 80 / distance.
Step 7: Solve for distance. Rearranging gives distance = height / tan(60°).
Step 8: Calculate tan(60°). The value of tan(60°) is √3.
Step 9: Substitute tan(60°) into the equation: distance = 80 / √3.
Step 10: Simplify the equation. To make it easier, multiply the numerator and denominator by √3: distance = (80√3) / 3.
Step 11: Final answer. The distance from the base of the hill to the car is 40√3 meters.

Trigonometry – The problem involves using the tangent function to relate the height of the hill and the distance to the car based on the angle of depression.
Angle of Depression – Understanding that the angle of depression from the top of the hill to the car is equal to the angle of elevation from the car to the top of the hill.
Right Triangle Properties – The scenario can be visualized as a right triangle where the height of the hill is one leg, the distance from the base of the hill to the car is the other leg, and the line of sight is the hypotenuse.