From the top of a tree the angle of depression of a boat is 30degree if the height of a tree is 36m above the sea level calculate the horizontal distance of the boat from the bottom of a tree?
I assume you made a sketch
In mine, 36/x = tan30°
x = 36/tan30 , but I know from my basic 30-60-90 triangle that tan30 = 1/√3
x = 36/(1/√3)
= 36√3 m or appr 62.35 m
To calculate the horizontal distance of the boat from the bottom of the tree, we can use trigonometry and the information given in the problem.
First, let's identify the given information:
- Angle of depression: 30 degrees
- Height of the tree: 36 meters
Now, let's define the variables:
- Let 'd' represent the horizontal distance from the bottom of the tree to the boat.
- Let 'h' represent the height of the boat above the sea level.
Given that the height of the tree is 36 meters above sea level, and the angle of depression is 30 degrees, we can set up the following trigonometric relationship:
tan(30 degrees) = opposite/adjacent
tan(30) = h/d
Next, let's solve for 'd' by rearranging the equation:
d = h / tan(30 degrees)
Since the height of the tree is 36 meters above sea level, the height of the boat (h) would be the sum of the height of the tree and the height above sea level:
h = 36 + 0 (since the boat is at sea level)
Substituting the value of 'h' into the equation, we get:
d = (36 + 0) / tan(30 degrees)
Now, we can calculate the horizontal distance by substituting the values into the equation and finding the tangent of 30 degrees:
d = 36 / tan(30 degrees)
Using a scientific calculator, we find that tan(30 degrees) is approximately 0.577.
d ≈ 36 / 0.577
Calculating the above expression, we find:
d ≈ 62.43 meters
Therefore, the horizontal distance of the boat from the bottom of the tree is approximately 62.43 meters.