A tree is 15 meters tall. From a point on the ground, the angle of elevation to
Practice Questions
Q1
A tree is 15 meters tall. From a point on the ground, the angle of elevation to the top of the tree is 30 degrees. How far is the point from the base of the tree?
15√3 meters
30 meters
45 meters
10 meters
Questions & Step-by-Step Solutions
A tree is 15 meters tall. From a point on the ground, the angle of elevation to the top of the tree is 30 degrees. How far is the point from the base of the tree?
Step 1: Understand the problem. We have a tree that is 15 meters tall and we want to find out how far away a point on the ground is from the base of the tree.
Step 2: Identify the angle of elevation. The angle of elevation to the top of the tree is given as 30 degrees.
Step 3: Recall the relationship between height, distance, and angle in a right triangle. We can use the tangent function, which is defined as the opposite side (height of the tree) over the adjacent side (distance from the tree).
Step 4: Write the formula for distance. The formula we will use is: Distance = height / tan(angle).
Step 5: Plug in the values. The height of the tree is 15 meters and the angle is 30 degrees. So, we have: Distance = 15 / tan(30°).
Step 6: Calculate tan(30°). The value of tan(30°) is √3 / 3.
Step 7: Substitute tan(30°) into the formula. Now we have: Distance = 15 / (√3 / 3).
Step 8: Simplify the equation. This is the same as multiplying by the reciprocal: Distance = 15 * (3 / √3).
Step 9: Calculate the distance. This simplifies to Distance = 15 * (3 / √3) = 15√3 meters.
Step 10: State the final answer. The distance from the point on the ground to the base of the tree is 15√3 meters.
Trigonometry – The problem involves using the tangent function to relate the height of the tree to the distance from the point on the ground.
Angle of Elevation – Understanding the angle of elevation is crucial for setting up the right triangle in the problem.
Right Triangle Properties – The relationship between the sides of a right triangle and the angles is fundamental to solving the problem.
