I am having a little trouble figuring this problem out if someone could help I would appreciate it. Question: The angle of elevation to the top f a tree sighted from ground level is 19degrees. If the observer moves 60 ft closer, the angle of elevation from ground level to the top of the tree is 37degrees. Find the height of the tree. Round intermediate calculations t for decimal places and final answers to one decimal place. What is the height of the tree. Thank you.
if you review the definition of the cotangent function, then you can see that with a height of h,
h cot19° - h cot37° = 60
To solve this problem, we can use the concept of trigonometry. Let's start by labeling the unknown values:
Let:
- h be the height of the tree
- x be the distance from the observer to the base of the tree
We are given that the angle of elevation to the top of the tree from the ground level is 19 degrees. This means that the tangent of the angle (tan(19)) is equal to the opposite side (height of the tree, h) divided by the adjacent side (distance from the observer to the base of the tree, x).
So, the initial equation is:
tan(19) = h / x ----(1)
Next, we are given that the observer moves 60 ft closer to the tree. This means the new distance from the observer to the base of the tree is (x - 60) ft. We are also given that the angle of elevation from ground level to the top of the tree is now 37 degrees. Using similar logic, we can write another equation:
tan(37) = h / (x - 60) ----(2)
Now, we have a system of equations with two unknowns (h and x). We need to solve for the height of the tree (h).
To do this, we can solve these two equations simultaneously. We can start by rearranging equation (1) to solve for x:
x = h / tan(19) ----(3)
Next, substitute equation (3) into equation (2):
tan(37) = h / (h / tan(19) - 60)
Simplify this equation by multiplying both sides by (h / tan(19) - 60):
tan(37) * (h / tan(19) - 60) = h
Expand this equation:
h * (tan(37) / tan(19)) - 60 * tan(37) = h
Move all the h terms to one side:
h * (tan(37) / tan(19)) - h = 60 * tan(37)
Factor out the h:
h * (tan(37) / tan(19) - 1) = 60 * tan(37)
Divide both sides by (tan(37) / tan(19) - 1):
h = (60 * tan(37)) / (tan(37) / tan(19) - 1)
Now, we can use a calculator to evaluate this expression.
Plugging in the values and rounding intermediate calculations to 4 decimal places, you will find that the height of the tree is approximately 113.5 ft (rounded to one decimal place).
Therefore, the height of the tree is approximately 113.5 ft.