Two people standing 30 feet apart look east at the top of a tree. The first person looks up at an angle of 78 degrees. The second person looks up at an angle of 62 degrees. How tall is the tree (to the nearest foot)?
Refer to the diagram:
http://img3.imageshack.us/i/11260110467.png/
H = height of the tree
x = distance of tree from first observer in feet
x+30 = distance of tree from second observer in feet
h=average height of observers' eys from ground in feet.
Using the trigonometric ratio
tan(θ)=opposite/adjacent, we have
tan(78°)=H/x
H=x*tan(78°).....(1)
tan(62°)=H/(x+30)
H=(x+30)*tan(62°).....(2)
Equate H from (1) and (2)
x*tan(78°) = (x+30)*tan(62°)
Solve for x.
Substitute the value of x into (1) to get H.
Finally, add 5 feet to H for the average height of eyes above ground.
I get 99 feet including adjustment for distance from ground to observers' eyes.... quite a tree!
Did you make a diagram?
I labeled the two people as P and Q, with Q closer to the tree.
I labeled the top of the tree T and its base O
It is easy to see that angle PTO = 16º
by sine law:
QT/sin62º = 30/sin16º
So you can find QT, which is the hypotenuse of triangle TOQ
then sin 78º = TO/QT
TO = QTsin78º
evaluate.
To find the height of the tree, we can use the concept of trigonometry and the given angles. Let's break down the problem:
1. Draw a diagram: Visualize two people standing 30 feet apart, with a tree between them. Label the distance between the two people as 30 feet. The first person looks up at an angle of 78 degrees, and the second person looks up at an angle of 62 degrees. Label the height of the tree as 'h.'
2. Observe the right triangles formed: The first person forms a right triangle with the tree, where the height of the tree is the opposite side and the distance between the two people is the adjacent side. The second person forms a similar right triangle.
3. Identify the trigonometric ratios: We can use the tangent function (opposite/adjacent) to relate the angles and sides of a right triangle.
For the first person:
tan(78 degrees) = h/30 feet
For the second person:
tan(62 degrees) = h/30 feet
4. Solve for the height of the tree 'h': Rearrange the equations to isolate 'h':
For the first person:
h = tan(78 degrees) * 30 feet
For the second person:
h = tan(62 degrees) * 30 feet
5. Calculate the height of the tree: Plug the values into a calculator or use a computing device to evaluate the trigonometric functions.
For the first person:
h = tan(78 degrees) * 30 feet
≈ 2.753 * 30 feet
≈ 82.59 feet
For the second person:
h = tan(62 degrees) * 30 feet
≈ 1.881 * 30 feet
≈ 56.44 feet
6. Determine the average height: Since the tree should have the same height, take the average of the two calculated heights:
Average height = (82.59 feet + 56.44 feet)/2
≈ 139.03 feet
Therefore, the height of the tree, to the nearest foot, is approximately 139 feet.