From the foot of a building i have to look upwards at an angle of 22 degrees 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 degrees to look at the top of the tree. A) How tall is the tree? B) How far from the building is the tree?
Please help me
To find the height of the tree and the distance from the building, we can use trigonometry. Let's break down the problem step-by-step:
Step 1: Determine the height of the building.
Given that the top of the building is 150 meters above the ground, the height of the building is 150 meters.
Step 2: Determine the distance from the foot of the building to the tree.
This distance will help us find the height of the tree using trigonometry. We'll call this distance "x".
Step 3: Determine the height of the tree using the angle of elevation.
Using the angle of elevation of 22 degrees, we can set up the following trigonometric equation:
tan(22 degrees) = height of the tree / distance from the building to the tree (x)
This equation can be rearranged to find the height of the tree:
height of the tree = x * tan(22 degrees)
Step 4: Determine the height of the tree using the angle of depression.
Using the angle of depression of 50 degrees, we can set up the following trigonometric equation:
tan(50 degrees) = height of the tree / height of the building (150 meters)
This equation can be rearranged to find the height of the tree:
height of the tree = 150 meters * tan(50 degrees)
Step 5: Set up an equation using the two heights of the tree.
Since the height of the tree is the same in both equations, we can set them equal to each other:
x * tan(22 degrees) = 150 meters * tan(50 degrees)
Step 6: Solve the equation to find the distance from the building to the tree (x).
Using trigonometric functions, we can solve the equation:
x = (150 meters * tan(50 degrees)) / tan(22 degrees)
Step 7: Calculate the height of the tree using the solved distance (x).
Using the equation from Step 3, we can substitute the value of x to find the height of the tree.
So, to summarize:
A) The height of the tree can be found using the equation: height of the tree = 150 meters * tan(50 degrees)
B) The distance from the building to the tree can be found using the equation: x = (150 meters * tan(50 degrees)) / tan(22 degrees)
To find the height of the tree and the distance from the building, we can use trigonometry.
Let's begin with part A: finding the height of the tree.
1) Draw a diagram to visualize the situation. Represent the building as a straight line, the tree as another line perpendicular to the ground, and the angles of elevation and depression.
^
|
T |------> Tree (Height = ?)
|
|
|
|------> Building (Height = 150m)
2) In this situation, we have a right-angled triangle. The height of the building is the opposite side, and the distance from the foot of the building to the tree is the adjacent side.
3) We know that the angle of elevation is 22 degrees, so we can label it as such.
4) Now, using the tangent function, we can set up the equation: tan(angle) = opposite/adjacent.
tan(22 degrees) = Tree height / 150 meters
Rearrange the equation to solve for the height of the tree:
Tree height = tan(22 degrees) * 150 meters
5) Use a calculator to determine the value of tan(22 degrees) and multiply it by 150 meters to get the height of the tree.
Now, let's move on to Part B: finding the distance from the building to the tree.
1) Draw a new diagram to visualize the situation, just as before. Label the height of the tree as the opposite side, and the distance from the building to the tree as the adjacent side.
2) We know that the angle of depression is 50 degrees, so we can label it as such.
3) Now, using the tangent function again, we can set up the equation: tan(angle) = opposite/adjacent.
tan(50 degrees) = Tree height / distance from the building to the tree
Rearrange the equation to solve for the distance from the building to the tree:
Distance from the building to the tree = Tree height / tan(50 degrees)
4) Use the height of the tree that was previously calculated and divide it by the value of tan(50 degrees) to find the distance from the building to the tree.
Using these calculations, you will be able to find the height of the tree as well as the distance from the building to the tree.
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