A surveyor is standing 550 feet from the base of a redwood tree in the California Redwood Forest. The tree is 362 feet tall. Sketch a drawing that models the situation and then find the angle of elevation of the top of the tree from the spot where the surveyor is standing.
ThetaElevation= arcTan(362/550)
To sketch a drawing that models the situation, follow these steps:
1. Draw a horizontal line to represent the ground.
2. Mark one end of the line as the base of the redwood tree.
3. Measure and mark a vertical line above the base to represent the height of the tree. This line should be 362 feet long.
4. From the other end of the horizontal line (representing the spot where the surveyor is standing), draw a diagonal line connecting to the top of the tree.
5. Label the horizontal line as "Ground," the vertical line as "Tree Height," and the diagonal line as "Surveyor's line of sight."
6. Indicate the distance from the surveyor to the tree by drawing a line segment connecting the surveyor's location to the base of the tree. This line should be 550 feet long.
To find the angle of elevation of the top of the tree, use trigonometry. The tangent function can be used in this case.
Let θ be the angle of elevation.
Tangent(θ) = Opposite/Adjacent = Tree Height/Distance between surveyor and tree
Tangent(θ) = 362/550
Now, we can find θ by taking the inverse tangent (arctan) of both sides.
θ = arctan(362/550)
Using a calculator, we can find the approximate value of this angle.
To find the angle of elevation of the top of the tree from the spot where the surveyor is standing, we can use basic trigonometry.
First, let's sketch a diagram to visualize the situation:
```
|
| *
| /|
| / |
| / |
550 ft | / | 362 ft
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
| / |
|/____________________________|
```
In the diagram, the surveyor is standing 550 feet away from the base of the redwood tree, which is 362 feet tall. We need to find the angle of elevation of the top of the tree, indicated by the asterisk (*).
To find this angle, we can use the tangent function:
tan(θ) = opposite/adjacent
In this case, the opposite side is the height of the tree (362 ft), and the adjacent side is the distance from the surveyor to the base of the tree (550 ft). Therefore:
tan(θ) = 362/550
To find θ, we can take the inverse tangent (also known as arctan) of both sides:
θ = arctan(362/550)
Using a calculator, we find that:
θ ≈ 33.85 degrees
So, the angle of elevation of the top of the tree from the spot where the surveyor is standing is approximately 33.85 degrees.
Using a scientific calculator or graphing calculator to find the inverse tangent of the ratio. Round to the nearest degree.
4072-06-02-03-00_files/i0280000.jpg
a.
36°
b.
54°
c.
46°
d.
44°