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From a point on the ground, the angle of elevation of the top of a hill is 30 de

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Question: From a point on the ground, the angle of elevation of the top of a hill is 30 degrees. If the height of the hill is 50 meters, how far is the point from the base of the hill?

Options:

  1. 50 m
  2. 75 m
  3. 100 m
  4. 125 m

Correct Answer: 100 m

Solution: 

Using tan(30°) = height/distance, we have 1/√3 = 50/distance. Therefore, distance = 50√3 ≈ 86.6 m.

From a point on the ground, the angle of elevation of the top of a hill is 30 de

Practice Questions

Q1
From a point on the ground, the angle of elevation of the top of a hill is 30 degrees. If the height of the hill is 50 meters, how far is the point from the base of the hill?
  1. 50 m
  2. 75 m
  3. 100 m
  4. 125 m

Questions & Step-by-Step Solutions

From a point on the ground, the angle of elevation of the top of a hill is 30 degrees. If the height of the hill is 50 meters, how far is the point from the base of the hill?
  • Step 1: Understand the problem. We have a hill that is 50 meters tall, and we want to find out how far away we are from the base of the hill when we look up at it at an angle of 30 degrees.
  • Step 2: Recall the relationship between the angle of elevation, height, and distance. We can use the tangent function, which is defined as the opposite side (height of the hill) over the adjacent side (distance from the base).
  • Step 3: Write down the formula for tangent: tan(angle) = height / distance.
  • Step 4: Substitute the known values into the formula. We have tan(30°) = height (50 meters) / distance.
  • Step 5: Find the value of tan(30°). It is equal to 1/√3.
  • Step 6: Set up the equation: 1/√3 = 50 / distance.
  • Step 7: Rearrange the equation to solve for distance: distance = 50 * √3.
  • Step 8: Calculate the distance. Using a calculator, 50 * √3 is approximately 86.6 meters.
  • Step 9: Conclude that the point is about 86.6 meters away from the base of the hill.
  • Trigonometry – The problem tests the understanding of the tangent function in right triangles, specifically how to relate angles and side lengths.
  • Angle of Elevation – The question involves interpreting the angle of elevation from a horizontal line to the top of an object (the hill).
  • Right Triangle Properties – The problem requires knowledge of the properties of right triangles to solve for the distance using the height and angle.
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