1. A rangers tower is located 44m from a tall tree. From the top of the tower, the angle of elevation to the top of the tree is 28 degrees and the angle of depression to the base is 36 degrees. How tall is the tree?
2. An engineer determines that the angle of elevation from her position to the top of the tower is 52 degrees. She measures the angle of elevation again from the point 47m far from the tower and finds it to be 31 degrees. Both positions are due east of the town. Find the height of the tower.
1.
height of tree above her line of sight:
h1/44 = tan28
h1 = 44tan28
height of tree below her line of sight:
h2/44 = tan36
h2 = 44tan36
height of tree = h1 + h2 = ...
(not a good idea to have her view to be below the tree line)
2. make a sketch. Label the top of the tree T, her first position B and her second position A
angle A = 31
angle TBA = 128
then angle BTA = 21°
At this point I suspect a typo. The usual distance given for this type of question is the distance between here two observation points.
The way you have it, we could simply do:
height of tower/47 = tan 52
height of tower = 47tan52 = ....
and all the other stuff would not be needed.
Before I proceed , please confirm which way the wording should be.
83.95
in Lenin Square.
yea
1. Well, well, well! Looks like we have some angular fun going on here! Let's break it down like a comedian working a crowd. First, let's find the height of the tower using the angle of depression.
If the angle of depression is 36 degrees, then the angle of elevation to the top of the tree is its complementary angle: 180 degrees - 36 degrees = 144 degrees.
Next, we can use some trigonometry wizardry. We have a right triangle, with the distance from the tower to the tree as the horizontal leg, the height of the tower as the vertical leg, and the angle of elevation as the angle between them.
Using the tangent function, we can set up this equation:
tan(144 degrees) = Height of the tree / 44m
But wait, we need to find the height of the tree, not the tangent! Silly me! Rearranging the equation, we get:
Height of the tree = 44m * tan(144 degrees)
Now it's just a matter of plugging in the values and calculating the height. Remember, math is like a joke - it's all about the delivery!
2. Ah, another angle-loving problem! This time, we have two positions from which our engineer measures the angle of elevation to the top of the tower. Let's put on our comedy hats and solve it!
We have two positions, so let's call the distance from the first position "x." From the first position, the angle of elevation is 52 degrees. From the second position, the angle of elevation is 31 degrees and the distance from the tower is 47m.
Using some triangle humor, we can set up two equations.
For the first position:
tan(52 degrees) = Height of the tower / x
For the second position:
tan(31 degrees) = Height of the tower / (x + 47m)
Now it's time for some hilarious algebra!
We can rearrange the first equation to find the height of the tower:
Height of the tower = x * tan(52 degrees)
And for the second equation:
Height of the tower = (x + 47m) * tan(31 degrees)
Since both equations represent the same height, we can set them equal to each other and solve for x. Once we find the value of x, we can substitute it back into either equation to find the height of the tower. Remember, math can be funny too!
To solve these problems, we can use basic trigonometry principles, specifically the tangent function.
1. Let's first consider the problem about the ranger's tower and the tall tree.
From the top of the tower, the ranger looks up to the top of the tree. This forms an angle of elevation of 28 degrees.
Drawing a diagram, you can visualize a right-angled triangle with the tower, the top of the tree, and the ranger at the top vertex.
The side opposite the 28-degree angle is the height of the tree (h), and the adjacent side is the distance between the tower and the tree (44m).
Using the tangent function, we have the equation:
tan(28°) = h/44
To find h, multiply both sides of the equation by 44:
h = 44 * tan(28°)
Using a calculator, we can find:
h ≈ 21.7 meters
Therefore, the tree is approximately 21.7 meters tall.
2. Now let's solve the problem with the engineer and the tower.
The engineer measures the angle of elevation to the top of the tower from two different points, both due east of the town.
We have two right-angled triangles, one from each location.
In the first triangle, the engineer is directly east of the tower, and the angle of elevation is 52 degrees.
In the second triangle, the engineer is 47 meters east of the tower, and the angle of elevation is 31 degrees.
Let's denote the height of the tower as h and the distance from the first point to the tower as x. Therefore, the distance from the second point to the tower is (x + 47)m.
Using the tangent function, we can set up two separate equations:
tan(52°) = h/x
tan(31°) = h/(x + 47)
We can now solve this system of equations to find the value of h.
Dividing the second equation by the first equation, we eliminate h:
tan(31°) / tan(52°) = (h/(x + 47)) / (h/x)
tan(31°) / tan(52°) = (x / (x + 47))
Now, we can solve for x. Rearranging the equation:
x / (x + 47) = tan(31°) / tan(52°)
Cross-multiplying:
x * tan(52°) = (x + 47) * tan(31°)
Expanding:
x * 0.6658 = (x + 47) * 0.6009
Simplifying the equation:
0.6658x = 0.6009x + 28.2483
0.065x = 28.2483
x ≈ 434.6
Substituting this value back into either of the original equations (e.g., x/tan(52°) = h), we can find the height of the tower.
Using a calculator, we can solve for h:
h ≈ 434.6 / tan(52°)
h ≈ 434.6 / 1.2799
h ≈ 339.8 meters
Thus, the height of the tower is approximately 339.8 meters.