The top of a building 24 meter high is observed from the top and from the bottom of the vertical tree. The angle of elevation are found to be 45 degree and 60 degree respectively. Find the height of the tree
Let's assume the height of the tree is h meters.
From the top of the building, the angle of elevation is 45 degrees, meaning the observer's line of sight forms a 45-degree angle with the horizontal. This forms a right triangle with the building height as the opposite side and the distance from the tree to the building as the adjacent side.
Using the trigonometric ratio for tangent, we have:
tan(45°) = building height / distance from the tree to the building
1 = 24 / distance from the tree to the building
distance from the tree to the building = 24 meters
From the bottom of the tree, the angle of elevation is 60 degrees, meaning the observer's line of sight forms a 60-degree angle with the horizontal. This forms a right triangle with the height of the tree as the opposite side and the distance from the tree to the building as the adjacent side.
Using the trigonometric ratio for tangent, we have:
tan(60°) = height of the tree / distance from the tree to the building
√3 = h / 24
h = 24√3 meters
Therefore, the height of the tree is approximately 41.57 meters.
To find the height of the tree, we can use the concept of trigonometry.
Let's assume the height of the tree is "h."
From the top of the tree, the angle of elevation is 45 degrees, and from the bottom of the tree, the angle of elevation is 60 degrees.
Based on this information, we can create two right-angled triangles.
First, let's consider the triangle formed by the observer at the top of the building and the top of the tree:
In this triangle, the side opposite the angle of elevation of 45 degrees is the height of the building (24 meters), and the side opposite the right angle is the height of the tree (h).
Using the trigonometric ratio for the tangent of an angle:
tan(45 degrees) = opposite/adjacent
tan(45 degrees) = h/24
Simplifying this equation, we get:
1 = h/24
h = 24 meters
Now, let's consider the triangle formed by the observer at the bottom of the tree and the top of the tree:
In this triangle, the side opposite the angle of elevation of 60 degrees is the height of the building (24 meters + h), and the side opposite the right angle is the height of the tree (h).
Using the trigonometric ratio for the tangent of an angle:
tan(60 degrees) = opposite/adjacent
tan(60 degrees) = h/(24 + h)
Simplifying this equation, we get:
√3 = h/(24 + h)
Now, we can solve this equation to find the value of h:
√3(24 + h) = h
√3 * 24 + √3 * h = h
√3 * 24 = h - √3 * h
24√3 = h(1 - √3)
h = 24√3 / (1 - √3)
h ≈ 41.6 meters
Therefore, the height of the tree is approximately 41.6 meters.
To find the height of the tree, we can use the concept of trigonometry. Let's denote the height of the tree as 'h'.
From the top of the vertical tree, the top of the building is observed at an angle of elevation of 45 degrees. This means that if we draw a right-angled triangle, the angle between the ground and the line of sight to the top of the building is 45 degrees.
From the bottom of the vertical tree, the top of the building is observed at an angle of elevation of 60 degrees. Again, drawing a right-angled triangle, we can see that the angle between the ground and the line of sight to the top of the building is 60 degrees.
Now, let's calculate the height of the tree using trigonometric ratios.
In the triangle formed from the top of the tree, the height of the building is the opposite side and the distance from the tree to the base of the building is the adjacent side. Since the angle is 45 degrees, we use the tangent function:
tan(45) = opposite/adjacent
1 = h/24
h = 24 meters
In the triangle formed from the bottom of the tree, the height of the building is again the opposite side and the distance from the tree to the base of the building is the adjacent side. Since the angle is 60 degrees, we use the tangent function:
tan(60) = opposite/adjacent
√3 = h/x (since 1 / √3 = x / h)
x = h / √3 = 24 / √3 ≈ 13.85 meters
Therefore, the height of the tree is approximately 13.85 meters.