From lookout tower 80ft. high, a man observes from a position 6.5 ft. below the top of the tower that the angle of elevation of the top of the certain tree is 12deg40mins and the angle of depression of its base is 70deg20mins. If the base oof the tower and the base of the tree are the same level, what is the height of the tree?
I will help you with the diagram, after that it is easy.
Draw a vertical line AB, the tower, where A is the position of the man and B is the base of the tower.
AB = 73.5
Draw another vertical line CD, for the tree, C as the top and D as the bottom
Mark E on CD as the same height as AB
clearly AE = BD
and ED = AB , all we need is CE
angle (CAE) = 12.66667°
angle(EAD) = 70.3333° making angle (BAD) = 19.66667°
You now have 2 nice right-angled triangles, where you can find BD, and its an easy ride after that
To find the height of the tree, we can use trigonometry and set up a right triangle.
First, let's draw a diagram to visualize the situation:
```
A
|\
| \
| \
| \ h*
| \
-----|-----\
/ 80ft \
/ \
B / \ C
```
Here, A represents the top of the tree, B represents the base of the tree, and C represents the base of the lookout tower.
The height of the lookout tower is given as 80ft. Let's call the height of the tree as "h".
Now, we need to find the lengths AC (height of the tower) and AB (height of the tree) to determine h.
Let's assume x be the horizontal distance from the base of the tower to the tree.
To find AC (height of the tower), we can use the angle of elevation. Since we know that the man observes the top of the tree from a position 6.5 ft. below the top of the tower, the distance CA is 80ft - 6.5ft = 73.5ft.
Next, to find AB (height of the tree), we need to find the horizontal distance BC, and then use the angle of depression.
To find BC, we can use the trigonometric relationship:
tan(angle) = opposite/adjacent
In this case, tan(12deg 40mins) = BC / 73.5ft. We need to convert the angle to decimal form before calculating the tangent.
Angle in decimal form = 12 + 40/60 = 12.667 degrees
So, tan(12.667 degrees) ≈ BC / 73.5ft.
Now, let's solve this equation for BC:
BC ≈ tan(12.667 degrees) * 73.5ft
Using a calculator, we find BC ≈ 16.513ft.
Finally, to find AB, the height of the tree, we use the angle of depression:
tan(angle) = opposite/adjacent
In this case, tan(70deg 20mins) = AB / 73.5ft.
Angle in decimal form = 70 + 20/60 = 70.333 degrees
So, tan(70.333 degrees) ≈ AB / 73.5ft.
Now, let's solve this equation for AB:
AB ≈ tan(70.333 degrees) * 73.5ft
Using a calculator, we find AB ≈ 203.599ft.
Therefore, the height of the tree (h) is equal to AB, which is approximately 203.599ft.