After a hurricane, two homeowners examine the trees on their property for damage. They think a 20 ft palm tree is leaning slightly from its original vertical position. From a point of 23 ft away, they measure the angle of elevation to the top of the palm tree as 43 degrees. Is the palm tree leaning? If so, by how many degrees from the vertical? Round to the nearest tenth of a degree.
To determine if the palm tree is leaning, we can calculate the angle of deviation from the vertical position. Let's assume that the angle of deviation is α.
From the given information, we can form a right triangle with the base being the distance from the point of observation to the tree (23 ft), the height being the height of the tree (20 ft), and the hypotenuse being the distance between the point of observation and the top of the tree.
Using trigonometry, we know that the tangent of an angle is equal to the opposite side divided by the adjacent side.
In this case, tan(α) = height of the tree / distance to the tree
tan(α) = 20 ft / 23 ft
α = arctan(20/23)
Using a calculator, we find that α ≈ 40.96 degrees.
Since the angle of elevation to the top of the palm tree is 43 degrees and the angle of deviation is approximately 40.96 degrees, we can conclude that the palm tree is leaning. The tree is leaning by approximately 43 degrees - 40.96 degrees = 2.04 degrees from the vertical position. Rounded to the nearest tenth, the palm tree is leaning by 2.0 degrees from the vertical.
To determine if the palm tree is leaning and by how many degrees, we can use trigonometry.
First, let's draw a diagram to visualize the situation:
```
A
|\
| \
| \ 20 ft
| \
| \ .
| \
| \
|-------\ B
23 ft
```
In this triangle, point A represents the top of the palm tree, point B represents the point of measurement, the length of AB is 20 ft, and the length of BC is 23 ft.
We can use the tangent of the angle of elevation to find the height of the tree from point B:
tan(angle) = opposite/adjacent
tan(43 degrees) = AB/BC
tan(43 degrees) = height/23 ft
To solve for the height, we can rearrange the equation:
height = tan(43 degrees) * 23 ft
Using a calculator, we find:
height ≈ 24.89 ft
Since the palm tree is leaning, we need to find the angle at which it is leaning from the vertical. Let's call this angle x.
In the triangle formed by the vertical line, the tree trunk, and the leaning tree, the angle at the top of the triangle is 90 degrees, and angle x is the angle at the bottom. Since the sum of the angles in a triangle is 180 degrees, we can subtract 90 degrees and the angle of elevation (43 degrees) from 180 degrees to find angle x:
x = 180 degrees - 90 degrees - 43 degrees
x ≈ 47 degrees
Therefore, the palm tree is leaning by approximately 47 degrees from the vertical.
To determine whether the palm tree is leaning and by how many degrees, we can use trigonometry. Specifically, we'll use the tangent function, which relates the angle of elevation, the distance from the observer to the object, and the height of the object.
Let's define some variables to make our calculations easier:
- Let h be the height of the palm tree (the distance from the ground to the top).
- Let d be the distance between the homeowners and the palm tree.
We're given that the distance between the homeowners and the palm tree is 23 ft (d = 23 ft), and the angle of elevation to the top of the palm tree is 43 degrees.
Using the tangent function, we can write the following equation:
tan(angle) = opposite/adjacent
In this case, the opposite side is the height of the palm tree (h), and the adjacent side is the distance between the homeowners and the palm tree (d). Therefore, we have:
tan(43 degrees) = h/d
To find the height of the palm tree, we rearrange the equation and solve for h:
h = tan(43 degrees) * d
Plugging in the values, we have:
h = tan(43 degrees) * 23 ft
Using a scientific calculator or an online trigonometry calculator, we find that:
h ≈ 26.36 ft
If the palm tree were perfectly vertical, the height measured by the homeowners would be 20 ft. However, the actual height is approximately 26.36 ft, which means the tree is leaning.
To determine the degree to which the palm tree is leaning from the vertical, we need to find the difference between the angle of elevation to the top of the tree and 90 degrees (the angle of elevation for a perfectly vertical tree).
Let's calculate it:
Lean degrees = 90 degrees - angle of elevation
Lean degrees = 90 degrees - 43 degrees
Lean degrees ≈ 47 degrees
Therefore, the palm tree is leaning by approximately 47 degrees from the vertical.