frrom outlook tower 80ft. high, a man observes from a position 6.5ft. below the top of the tower that the angle of elevstion of the top of certain tree is 12deg40mins and the angle of depression of its base is 72deg20mins. If the base of the tower and the base of the tree are at the same level, what is the height of the tree?
The line of sight of the 72 deg angle
is represented by the hyp. of a rt triangle. The ht. of the observer above
the bottom of tree represents the ver.
side of the triangle.The hor. side is
the dist. from the tree to the bottom
of tower.The line of sight of the 12
deg. angle is the hyp of a 2nd rt.
triangle. Its' hor. side is = to hor.
side of 1st triangle.
Tan(72,20min) = 73.5 / X,
X = 73.5 / Tan(72,20min) = 23.41 Ft.
= hor. dist. from tree to tower.
Tan(12,40min) = Y / 23.41 ,
Y = 23.41 * Tan(12,40min) = 5.26 Ft
= ver. side of 2nd triangle.
h = 73.5 - 5.26 = 68.2 Ft. = ht.
of tree.
To find the height of the tree, we can use trigonometric concepts and apply them to the given information. Let's break down the problem.
1. We know that the man is observing the tree from a position 6.5 ft below the top of the tower. So, the height of the observation point is 80 ft - 6.5 ft = 73.5 ft.
2. We are given the angle of elevation of the top of the tree as 12°40'. This angle is formed between the horizontal line (from the observation point to the base of the tree) and the line from the observation point to the top of the tree.
3. We are also provided with the angle of depression of the base of the tree as 72°20'. This angle is formed between the horizontal line (from the observation point to the base of the tree) and the line from the observation point to the base of the tree.
We can use the tangent function to find the height of the tree.
Let's calculate the height of the tree step by step:
1. Find the distance from the observation point to the tree's base using the angle of depression:
tangent(72°20') = height of the tree / distance to the tree's base
Let's call the distance to the tree's base 'x'.
Using the tangent function, we have:
tan(72°20') = height of the tree / x
2. Find the distance from the observation point to the tree's top using the angle of elevation:
tangent(12°40') = height of the tree / (x + distance from the tree's base to the top of the tree)
Let's call the distance from the tree's base to the top of the tree 'y'.
Using the tangent function, we have:
tan(12°40') = height of the tree / (x + y)
3. We know that the base of the tower and the base of the tree are at the same level. So, x + y should be equal to the total height of the tower, which is 73.5 ft.
Now we can set up a system of equations to solve for x and y:
Equation 1: tan(72°20') = height of the tree / x
Equation 2: tan(12°40') = height of the tree / (x + y)
Equation 3: x + y = 73.5 ft
Solve this system of equations to find the values of x and y. Once you have the values, you can calculate the height of the tree by substituting them back into Equation 1 or Equation 2.