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?
If the tree is of height h and at distance d, then
(150-h)/d = tan 50°
h/d = tan 22°
Now you have two equations in two unknowns, so just solve them.
To find the height of the tree, we can use the tangent function since we have the angle and the opposite side (height).
a. Height of the tree:
Let's assume the height of the tree is H.
Using the tangent function:
tangent(angle) = opposite/adjacent
For the first scenario (from the foot of the building):
tan(22°) = H/adjacent
tan(22°) = H/x
For the second scenario (from the top of the building):
tan(50°) = H/150
tan(50°) = H/150
Now we have two equations:
1. tan(22°) = H/x
2. tan(50°) = H/150
To solve these equations, we can rearrange them to find the value of H.
For the first equation, we can rearrange it to solve for x:
x = H/tan(22°)
Substitute this value of x into the second equation:
tan(50°) = H/150
tan(50°) = H/(H/tan(22°))
tan(50°) = (H * tan(22°))/H
Hence, H cancels out and we are left with:
tan(50°) = tan(22°)
Solving this equation, we find that the height of the tree is the same in both scenarios, as the angles are the same:
H = x * tan(22°)
H = 150 * tan(22°)
Now we can calculate the height of the tree using the mathematical value of tan(22°).
For b, to find the distance from the building to the tree, we can use the tan function again and the angle of depression.
b. Distance from the building to the tree:
Let's assume the distance from the building to the tree is D.
Using the tangent function:
tangent(angle) = opposite/adjacent
In this case, the angle is the angle of depression of 50°, and the opposite side is the height of the tree, H.
tan(50°) = H/D
Rearranging the equation, we can solve for D:
D = H/tan(50°)
Substituting the value of H that we calculated earlier, we can find the distance D using the mathematical value of tan(50°).