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 the building, 150m above ground level, I have to look down at an angle of 50 degrees below the horizontal to sigh the tree top.
a.) How hight is the tree?
b.) How far from the building is this tree?
Let X be the distance from building t tree. H = tree height. Solve this set of equations:
H/X = tan 22
(150-H)/X = tan 50
(150-H)/H = tan 50/tan 22
This can be solved for H, using algebra and trig functions. Once you know H, solve for X.
h=94.3
x=37.7
To solve this problem, we can use trigonometry.
a.) To find the height of the tree, we can use the tangent function. Let's represent the height of the tree as 'h'.
From the foot of the building, we have an angle of 22 degrees. The tangent of an angle is equal to the opposite side divided by the adjacent side.
So, tan(22) = h / x (where 'x' is the distance from the building to the tree)
To solve for 'h', we rearrange the equation:
h = x * tan(22)
b.) To find the distance from the building to the tree, we can use the sine or cosine function. Let's represent the distance as 'd'.
From the top of the building, we have an angle of 50 degrees below the horizontal. The sine of an angle is equal to the opposite side divided by the hypotenuse (in this case, the height of the tree plus the height of the building).
So, sin(50) = h / (h + 150)
To solve for 'h', we rearrange the equation:
h = (h + 150) * sin(50)
After solving these equations, we can find both the height of the tree and the distance from the building to the tree.
Let's calculate:
First, we solve for 'h' using the second equation:
h = (h + 150) * sin(50)
h = 150 * sin(50)
h ≈ 114.52 meters
Next, we plug the value of 'h' into the first equation to find 'x':
114.52 = x * tan(22)
x ≈ 304.82 meters
So, the height of the tree is approximately 114.52 meters, and the distance from the building to the tree is approximately 304.82 meters.
To solve this problem, we can apply trigonometry principles, specifically the tangent function. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. We'll solve the problem step by step:
a.) To find the height of the tree, we'll use the information given in the first scenario. Let's assume the height of the tree is represented by 'h' meters.
First, we'll find the length of the adjacent side, which is the height of the building (150 meters). Then, using the tangent function, we can set up the equation:
tan(22 degrees) = h / 150
Now, we rearrange the equation to solve for 'h':
h = 150 * tan(22 degrees)
Using a scientific calculator, compute the value for tan(22 degrees) and multiply it by 150 to find the height of the tree (h).
b.) To determine the distance from the building to the tree, we'll utilize the information given in the second scenario. Let's assume the distance from the building to the tree is represented by 'd' meters.
In this case, we have a horizontal angle of 50 degrees below the horizontal. To solve for 'd', we'll again use the tangent function:
tan(50 degrees) = h / d
Rearrange the equation to solve for 'd':
d = h / tan(50 degrees)
Now, substitute the value of 'h' that we obtained from part (a) into the equation, and compute the value for tan(50 degrees) to find the distance 'd' from the building to the tree.
By following these steps, you can calculate the height of the tree (a) and the distance from the building to the tree (b). Always make sure your calculator is set to degrees mode when working with trigonometric functions.