A man standing in the corridor of an eight storey building sees the top of a tree at an angle of depression thirty degrees. If the tree is 160 meters tall and the man's eyes are 240 meters from the ground. Calculate the angle of depression of the foot of the tree
Draw a diagram. It should be clear that the distance d between the building and the tree can be found using
80/d = tan 30° = 1/√3
so, d = 80√3
Now you want to find θ such that
tanθ = 240/(80√3) = √3
...
To find the angle of depression of the foot of the tree, we need to use trigonometry.
Let's set up a right-angled triangle with the tree, the man's eyes, and the foot of the tree. The height of the tree (opposite side) is given as 160 meters, and the distance from the man's eyes to the ground (adjacent side) is given as 240 meters.
Now, we can use tangent, which is defined as the ratio of the opposite side to the adjacent side in a right triangle.
Tangent of an angle = Opposite side / Adjacent side
Let's call the angle of depression of the foot of the tree x.
Therefore, tangent(x) = opposite side / adjacent side
tangent(x) = 160 / 240
To find the angle x, we need to find the inverse tangent (also called arctan) of the tangent value.
x = arctan(160 / 240)
Using a calculator or trigonometry table, we can find x ≈ 33.69 degrees.
So, the angle of depression of the foot of the tree is approximately 33.69 degrees.