From the top of a hill, the angle of depression to a car parked on the road is 3
Practice Questions
Q1
From the top of a hill, the angle of depression to a car parked on the road is 30 degrees. If the height of the hill is 60 meters, how far is the car from the base of the hill?
60√3 meters
30√3 meters
90 meters
120 meters
Questions & Step-by-Step Solutions
From the top of a hill, the angle of depression to a car parked on the road is 30 degrees. If the height of the hill is 60 meters, how far is the car from the base of the hill?
Correct Answer: 60√3 meters
Step 1: Understand the problem. We have a hill that is 60 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 30 degrees.
Step 3: Visualize the situation. Imagine a right triangle where the height of the hill is one side (60 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 divided by the adjacent side. Here, the opposite side is the height of the hill (60 meters) and the adjacent side is the distance we want to find.
Step 5: Write the formula. We can express this as: tan(angle) = opposite / adjacent. For our case: tan(30 degrees) = height / distance.
Step 6: Substitute the known values. We know that tan(30 degrees) is 1/√3, the height is 60 meters, and we need to find the distance. So, we write: 1/√3 = 60 / distance.
Step 7: Rearrange the formula to find distance. Multiply both sides by distance: distance * (1/√3) = 60. Then, divide both sides by (1/√3): distance = 60 / (1/√3).
Step 8: Simplify the expression. To simplify 60 / (1/√3), we multiply by √3/√3: distance = 60√3 meters.
Angle of Depression – The angle formed by a horizontal line and the line of sight to an object below the horizontal line.
Trigonometric Ratios – Using tangent to relate the height of the hill and the distance to the car.
Right Triangle Properties – Understanding the relationship between the height, distance, and angles in a right triangle.
