From a building EF, the angles of depression to the top and bottom of a tree are 41° and 62° respectively. The tree is 15m from the building. (1)Draw diagram to show and (2) determine the height of the tree.
I assume you sketched the diagram
From the point of observation, draw a horizontal to the tree, that distance will be 15 metres.
You now have two right-angled triangles.
let the height of the tree be made up of h1 and h2
tan62° = h1/15 , h1 = 15tan62
similarly find h2
Your tree will have a height of
h1 + h2
= ....
Tan41 = h2/15.
h2 = 13 m.
Tan62 = (h1+h2)/15
h1+h2 = 28 m.
h1+13 = 28,
h1 = 15 m. = Ht. of tree.
To draw the diagram, follow these steps:
1. Draw a horizontal line to represent the ground and label it.
2. At one end of the ground line, draw a vertical line to represent the building and label it as EF.
3. At the other end of the ground line, draw a slanted line to represent the tree.
4. Label the bottom of the tree as point G.
5. Measure 15m from point E on the ground line and mark the point as H.
6. Draw a line segment connecting points EF and GH.
7. Label the angle of depression to the top of the tree as 41° and the angle of depression to the bottom of the tree as 62°.
8. Finally, label the height of the tree as x.
H G
|-----------|
/ |
EF / |
/ |
------/------------
41° 62°
Now, let's determine the height of the tree.
From the diagram, we can form a right triangle EGH, where EG represents the height of the tree.
Using trigonometry, we can determine EG as follows:
In triangle EGH,
- tan(62°) = EG / 15 (opposite/adjacent)
- tan(62°) * 15 = EG
- EG = 26.97 meters (approximately)
Therefore, the height of the tree is approximately 26.97 meters.
To draw the diagram, follow these steps:
1. Start by drawing the building EF as a vertical line. Label the bottom as E and the top as F.
E
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F
2. From point E, draw a line downward at an angle of 41°. Label the point where the line intersects the ground as G.
E
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F G
3. From point F, draw a line downward at an angle of 62°. Label the point where the line intersects the ground as H.
E
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F G
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H
4. Draw a straight line connecting points G and H. Label the point where this line intersects EF as T.
E
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--|--
-- | --
-- | --
-- | --
F G----T-----H
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Now let's determine the height of the tree:
1. In the right triangle GEH, the angle EGH is 90°.
2. The angle EGF is the difference between the angles of depression: 62° - 41° = 21°.
3. We can use the tangent function to find the height of the tree. The tangent of the angle EGF is equal to the opposite side (GH) divided by the adjacent side (EF).
tan(21°) = GH / EF
4. Since EF is given as 15m, we can solve for GH:
GH = tan(21°) * EF
GH = tan(21°) * 15
GH ≈ 5.77m
Therefore, the height of the tree is approximately 5.77 meters.