From the top of a hill, the angle of depression to a car parked on the ground is 60 degrees. If the height of the hill is 80 meters, how far is the car from the base of the hill?
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From the top of a hill, the angle of depression to a car parked on the ground is 60 degrees. If the height of the hill is 80 meters, how far is the car from the base of the hill?
Q: From the top of a hill, the angle of depression to a car parked on the ground is 60 degrees. If the height of the hill is 80 meters, how far is the car from the base of the hill?
Step 1: Understand the problem. We have a hill that is 80 meters tall, and we need to find out how far the car is from the base of the hill.
Step 2: Identify the angle of depression. The angle of depression from the top of the hill to the car is 60 degrees.
Step 3: Visualize the situation. Imagine a right triangle where the height of the hill is one side (80 meters), the distance from the base of the hill to the car is the other side, and the line of sight from the top of the hill to the car 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 hill) divided by the adjacent side (distance from the base to the car).
Step 5: Write the formula. We can express this as: tan(angle) = height / distance.
Step 6: Rearrange the formula to find distance. This gives us: distance = height / tan(angle).
Step 7: Substitute the values. We know the height is 80 meters and the angle is 60 degrees. So we need to find tan(60 degrees).
Step 8: Calculate tan(60 degrees). The value of tan(60 degrees) is √3.
Step 9: Plug in the values into the formula. This gives us: distance = 80 / √3.
Step 10: Simplify the expression. To make it easier, we can multiply the numerator and denominator by √3 to get: distance = 80√3 / 3.
Step 11: Final answer. The distance from the base of the hill to the car is 40√3 meters.
