From a point 5m. above the water surface, the angle of elevation of the top of a certain tree is 40°10' while the angle of depression to the foot of the tree is 63°20'. Find the height of the tree and the distance from the point of observation.
-Complete Solution Please.
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Tan63.33 = Y1/X = 5/X
X*Tan63.33 = 5
X = 5/Tan63.33 = 2.51 m = Distance from
the point of observation.
Tan40.17 = Y2/X = Y2/2.51
Y2 = 2.51*Tan40.17 = 2.12 m.
h = Y1+Y2 = 5 + 2.12 = 7.12 m.
To find the height of the tree and the distance from the point of observation, we can use trigonometric principles.
Let's start by labeling the relevant information:
- The height of the observer above the water surface: 5m
- The angle of elevation to the top of the tree: 40°10'
- The angle of depression to the foot of the tree: 63°20'
Now, let's solve for the height of the tree:
Step 1: Convert the angle measurements from degrees, minutes, and seconds (DMS) to decimal degrees.
In decimal degrees:
- Angle of elevation: 40.10°
- Angle of depression: 63.33°
Step 2: Use the tangent function to find the distance from the observer to the tree.
Using the angle of depression, we can set up the following equation:
tan(63.33°) = height of the tree / distance from the observer to the tree
Step 3: Solve for the distance from the observer to the tree.
distance from the observer to the tree = height of the tree / tan(63.33°)
Step 4: Substitute the value for the height of the tree.
Let's assume the height of the tree is "h" meters.
Therefore, distance from the observer to the tree = h / tan(63.33°)
Step 5: Use the tangent function to find the height of the tree.
Using the angle of elevation, we can set up the following equation:
tan(40.10°) = height of the tree / distance from the observer to the tree
Step 6: Solve for the height of the tree.
height of the tree = distance from the observer to the tree * tan(40.10°)
Now, let's substitute the values and calculate the height of the tree and the distance from the point of observation:
distance from the observer to the tree = h / tan(63.33°)
height of the tree = distance from the observer to the tree * tan(40.10°)
Once you substitute the values and perform the calculations, you will find the height of the tree and the distance from the point of observation.