From the foot of a building I have to look upwards at an angle of 22degrees 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 50degrees to look at the top of the tree.
a. How tall is the tree?
b. How far from the building is the tree?
draw a horizontal line from the top of the tree to the building to create 2 right-angled triangle
That line is common to both
So, set up a trig equation for each triangle using that common side.
You can then use substitution to get your answer.
To solve this problem, we'll use the concept of trigonometry. Let's break down the problem into two parts:
a. Finding the height of the tree:
We can use the tangent function to find the height of the tree. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side within a right triangle.
In this case, the opposite side is the height of the tree and the adjacent side is the distance from the foot of the building to the tree. Therefore, we have the following equation:
tan(22°) = height of tree / distance to tree
Since we don't know the distance to the tree, we'll use a variable, let's call it d.
tan(22°) = height of tree / d..............(Equation 1)
b. Finding the distance from the building to the tree:
Similar to the first part, we'll use the tangent function to find the distance to the tree. In this case, the opposite side is the height of the tree + the height of the building (150 meters), and the adjacent side is again the distance from the foot of the building to the tree. We have the following equation:
tan(50°) = (height of tree + 150) / distance to tree
tan(50°) = (height of tree + 150) / d..............(Equation 2)
To solve these equations simultaneously, we'll isolate the variables and use substitution:
From Equation 1, we get:
height of tree = d * tan(22°)
Now, substitute this expression into Equation 2:
tan(50°) = (d * tan(22°) + 150) / d
Simplifying, we have:
tan(50°) = tan(22°) + 150 / d
Cross-multiplying, we get:
d * tan(50°) = tan(22°) * d + 150
Rearranging the equation, we get:
d * tan(50°) - tan(22°) * d = 150
Factoring out the common term, we have:
d * (tan(50°) - tan(22°)) = 150
Finally, solving for d, we get:
d = 150 / (tan(50°) - tan(22°))
Once you have the value of d, you can substitute it back into Equation 1 to find the height of the tree:
height of the tree = d * tan(22°)
I hope this explanation helps you understand how to solve this problem using trigonometry.