The angle of elevation from a point on the ground to the top of a tree is 38 degrees. The angle of elevation from a point 22.5 ft farther back to the top of the tree is 28.2 degrees. Find the height of the tree to 2 decimal places.
could you help me out..I got this so far..thanks
h/(tan38(=h/(tan28.2)-29
Make a sketch.
On the ground line, mark off the bottom of the tree as Q, your first position
as A and your second point further back as B.
Label the top of the tree as P
So we want to find PQ
In triangle PBA, we know angle PBA = 28.2° and angle PAQ = 142°,
which make angle APB = 9.8°
by the sine law:
22.5/sin9.8 = AP/sin28.2°
AP = 22.5 sin28.2/sin 9.8 = 62.4665... ( I stored that in memory)
In the right-triangle APQ
sin 38 = PQ/AP
PQ = APsin38 = 62.4665..(sin38) = 38.458..
= 38.46 ft
To solve for the height of the tree, let's set up two ratios based on the given information.
Let h be the height of the tree and x be the distance from the first point to the tree.
From the first point:
tan(38°) = h / x
From the second point (which is 22.5 ft farther back):
tan(28.2°) = h / (x + 22.5)
Now we can solve for h by setting up a system of equations and solving simultaneously.
From the first point:
h = x * tan(38°) (equation 1)
From the second point:
h = (x + 22.5) * tan(28.2°) (equation 2)
Setting equation 1 equal to equation 2:
x * tan(38°) = (x + 22.5) * tan(28.2°)
Simplifying the equation:
x * tan(38°) = x * tan(28.2°) + 22.5 * tan(28.2°)
x * tan(38°) - x * tan(28.2°) = 22.5 * tan(28.2°)
Factoring out x:
x * (tan(38°) - tan(28.2°)) = 22.5 * tan(28.2°)
Dividing both sides by (tan(38°) - tan(28.2°)):
x = (22.5 * tan(28.2°)) / (tan(38°) - tan(28.2°))
Now that we have the value of x, we can substitute it back into equation 1 to find h:
h = x * tan(38°)
Substitute the value of x into the equation:
h = [(22.5 * tan(28.2°)) / (tan(38°) - tan(28.2°))] * tan(38°)
Calculate the value of h using a calculator to get the precise height of the tree to 2 decimal places.
To solve this problem, you can use the concept of trigonometry and set up a system of equations.
Let's define the following variables:
- h as the height of the tree
- d as the distance from the original point on the ground to the base of the tree
- x as the additional distance from the second point to the base of the tree
Now, using the given information, we can set up the following equations based on the angles of elevation:
Equation 1:
tan(38 degrees) = h / d
Equation 2:
tan(28.2 degrees) = h / (d + 22.5)
To find the height of the tree, we need to solve this system of equations.
First, let's rearrange Equation 1 to solve for d:
d = h / tan(38 degrees)
Now, substitute this expression for d in Equation 2:
tan(28.2 degrees) = h / (h / tan(38 degrees) + 22.5)
Next, simplify the equation:
tan(28.2 degrees) = tan(38 degrees) / (1 + 22.5 / (h * tan(38 degrees)))
Now, we can solve for h.
1. Multiply both sides of the equation by (h * tan(38 degrees))
tan(28.2 degrees) * (h * tan(38 degrees)) = tan(38 degrees)
2. Distribute the multiplication on the left side of the equation
tan(28.2 degrees) * h * tan(38 degrees) = tan(38 degrees)
3. Divide both sides of the equation by tan(28.2 degrees) * tan(38 degrees)
h = tan(38 degrees) / (tan(28.2 degrees) * tan(38 degrees))
Finally, you can calculate the value of h using a scientific calculator:
h = tan(38 degrees) / (tan(28.2 degrees) * tan(38 degrees))
After evaluating this expression, you will find the height of the tree. Remember to round the result to 2 decimal places as indicated in the question.