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?
How far from the building is the tree?
Assume the observations were made from the same building.
Let the distance of the building from the tree is D in m.
Then
D(tan22+tan50)=150
D=(150)/(tan22+tan50)
93.99
To find the height of the tree, we can use the concept of similar triangles. Let's break down the problem step by step.
Step 1: Finding the height of the building
Given that you are looking up at an angle of 22 degrees, we can set up a right triangle with the height of the building as the opposite side (let's call it h1) and the distance from your position to the building as the adjacent side. Since we don't have the actual distance, let's call it x.
Using trigonometry, we can write: tan(22) = h1 / x
Rearranging the equation, we have: h1 = x * tan(22)
Step 2: Finding the distance from the tree to the building
Now let's move on to finding the distance from the building to the tree. We will again set up a right triangle, but this time with the distance from the building as the opposite side (x) and the height of the tree as the adjacent side (let's call it h2).
Using trigonometry, we can write: tan(50) = h2 / x
Rearranging the equation, we have: h2 = x * tan(50)
Step 3: Setting up the final equation
Since the height of the building and the height of the tree together make up the total height (h) from your position, we can write: h = h1 + h2
Substituting the values of h1 and h2 from Step 1 and Step 2 into the equation, we get: h = x * tan(22) + x * tan(50)
Step 4: Solving for the height of the tree
Given that the height of the building is 150 meters, we can write the final equation as: h = 150 + x * tan(22) + x * tan(50)
Now we have a single equation with one unknown (x), so we can solve for x and then find the height of the tree.
To find the distance from the building to the tree, substitute the value of x back into any of the equations from Step 1 or Step 2.
Remember to always use the appropriate units for the calculations.