From a point on the ground, the angle of elevation to the top of a hill is 30 de
Practice Questions
Q1
From a point on the ground, the angle of elevation to the top of a hill is 30 degrees. If the distance from the point to the base of the hill is 100 meters, what is the height of the hill?
100√3 m
50 m
100 m
50√3 m
Questions & Step-by-Step Solutions
From a point on the ground, the angle of elevation to the top of a hill is 30 degrees. If the distance from the point to the base of the hill is 100 meters, what is the height of the hill?
Step 1: Understand that the angle of elevation is the angle formed between the horizontal ground and the line of sight to the top of the hill.
Step 2: Identify the distance from the point on the ground to the base of the hill, which is given as 100 meters.
Step 3: Recognize that we can use the tangent function (tan) to find the height of the hill. The formula is: tan(angle) = height / distance.
Step 4: Substitute the known values into the formula. Here, the angle is 30 degrees and the distance is 100 meters: tan(30°) = height / 100.
Step 5: Calculate tan(30°). The value of tan(30°) is 1/√3.
Step 6: Rewrite the equation using the value of tan(30°): 1/√3 = height / 100.
Step 7: To find the height, multiply both sides of the equation by 100: height = 100 * (1/√3).
Step 8: Simplify the expression: height = 100/√3.
Step 9: To get a numerical value, you can approximate 100/√3, which is about 57.74 meters, but the exact height is 100/√3.
Trigonometry – The problem tests the understanding of basic trigonometric functions, specifically the tangent function, which relates angles to opposite and adjacent sides in a right triangle.
Angle of Elevation – The question involves interpreting the angle of elevation from a point on the ground to the top of a hill, which is a common application of trigonometry in real-world scenarios.
Right Triangle Properties – The problem requires knowledge of the properties of right triangles, where the height of the hill represents the opposite side and the distance to the base represents the adjacent side.
