from a point A i observe that the angle of elevation of the top of the tree is 21 degree i walk 10m toward the foot of the tree to obtain B from where the angle of elevation of the top of the tree is 34 degree,calculate the height of the tree above the level of the observation
50 km? That's some tall tree. Did you draw a diagram? If so, it should be clear that the height h is found using
h cot21° - h cot34° = 10
h = 10/(cot21° - cot34°) = 8.9 m
To solve this problem, we can use trigonometry, specifically the tangent function. Here's how we can go about it:
Let's assume the height of the tree is "h" and the distance from point A to the foot of the tree is "x".
In right triangle ABC, where A is the observer's position, B is the new position after moving 10m towards the foot of the tree, and C is the top of the tree, we have two right angles at A and B.
Using the tangent function, we can set up the following equations:
tan(21°) = h / x (equation 1)
tan(34°) = h / (x + 10) (equation 2)
Now, we can solve these equations simultaneously to find the value of "h", which represents the height of the tree.
First, let's solve equation 1 for x:
x = h / tan(21°)
Now, substitute this value of x in equation 2:
tan(34°) = h / (h / tan(21°) + 10)
To simplify further, we can substitute tan(34°) ≈ 0.671 and tan(21°) ≈ 0.374 into the equation:
0.671 = h / (h / 0.374 + 10)
Now, we can solve for "h":
Multiply both sides by (h / 0.374 + 10):
0.671(h / 0.374 + 10) = h
Expand and simplify:
0.671(h) + 6.71 = h
Move all terms involving "h" to one side:
0.671h - h = -6.71
Combine like terms:
-0.329h = -6.71
Divide by -0.329 to isolate "h":
h = -6.71 / -0.329
Evaluate the expression:
h ≈ 20.4 meters
Therefore, the height of the tree above the level of observation is approximately 20.4 meters.