an observer and a tree are standing against a slope. the observer is 80m from the tree.the angle between the slope, from the feet of the observer,and thetop of the tree is 14 degree.the slope makes an angle of 9 degree with the horizontal. solve the triangle formed by the feet of the observer and the bottom and top of the tree.
To solve the triangle formed by the observer, the bottom of the tree, and the top of the tree, we can use trigonometric relationships.
Let's label the triangle:
The feet of the observer is point A.
The bottom of the tree is point B.
The top of the tree is point C.
We are given that AB = 80m and angle BAC = 14 degrees. We are also given that angle BAO (the slope angle) = 9 degrees.
First, let's find the height of the tree (AC) by finding the length of side BC. We can use trigonometry and the given angles.
Since AC is the opposite side of angle BAC, we can use the tangent function:
tan(angle BAC) = opposite/adjacent
tan(14 degrees) = BC/AB
BC = AB * tan(14 degrees)
BC = 80 * tan(14 degrees)
BC ≈ 20.32m
Now, let's find the distance along the slope from the observer to the bottom of the tree (BO). We can use trigonometry and the given slope angle.
Since BO is the opposite side of angle BAO, we can again use the tangent function:
tan(angle BAO) = opposite/adjacent
tan(9 degrees) = BO/AB
BO = AB * tan(9 degrees)
BO = 80 * tan(9 degrees)
BO ≈ 13.07m
Finally, we can find the height of the tree (AC) by subtracting BO from BC:
AC = BC - BO
AC ≈ 20.32m - 13.07m
AC ≈ 7.25m
Therefore, the height of the tree (AC) is approximately 7.25 meters.