From the foot of a building I have to look upwards at an angle of 22° to sight the top of a tree. From the top of a building, 150 meters above ground level, I have to look down at an angle of depression of 50° to look at the top of the tree.
a. How tall is the tree?
b. How far from the building is the tree?
assuming all these buildings are the same one, if the building is at distance d from the tree of height h,
draw a diagram to see that
(150-h)/d = tan 50°
h/d = tan 22°
plugging in the numbers,
h+1.19d = 150
h = 0.40d
So, we have
h = 94.3
d = 37.7
To solve this problem, we can use trigonometry, specifically the concepts of tangent and opposite angles. Let's break down each part of the problem separately:
a. How tall is the tree?
From the foot of the building, you're looking upwards at an angle of 22° to sight the top of the tree. Let's call the height of the tree "h".
Using trigonometry, we know that the tangent of an angle is the ratio of the opposite side to the adjacent side. In this case, the opposite side is "h" (the height of the tree) and the adjacent side is the distance from the foot of the building to the tree.
So, we have tan(22°) = h / x, where x is the unknown distance.
To solve for "h", we can rearrange the equation to h = x * tan(22°).
b. How far from the building is the tree?
From the top of the building, 150 meters above ground level, you're looking down at an angle of depression of 50° to view the top of the tree.
Again, using trigonometry, we can find the distance from the building to the tree. Let's call this distance "d".
Using the same concept of tangent, we can write the equation tan(50°) = h / (x + d), where h is the height of the tree and x is the unknown distance from the building to the tree.
Since we already know the value of h from part a, we can substitute it in the equation and solve for "d". It would be more convenient to rearrange the equation and solve for "d". The equation becomes:
d = (h / tan(50°)) - x.
Now, substitute the value of h (from part a) into the equation and solve for "d" using the given values.
Therefore, using trigonometry and these steps, you can find the height and distance of the tree from the building.