a boy standing 70m away from a flag–post observes that the angles of elevation of the top and bottom of the tower on top the flag–post are 70^ and 69^ respectively. Find the height of the tower
It appears that the flag-pole is positioned on a hill or on elevated ground in reference to the boy.
So is the 70 m the distance along the slant or is it the horizontal distance ?
I will assume it is the horizontal distance
Make your sketch.
1. Horizontal height to base of flagpole:
tan69 = x/70
x = 70tan69
2. Horizontal height to top of the flagpole
tan 70 = y/70
y = 70tan70
height of pole = y - x
= ....
The question changes totally if the slant distance is 70 m
To find the height of the tower, we can use the concept of trigonometry and set up a mathematical equation based on the given information.
Let's consider the following diagram:
Flag
|
| /|
| / | h
| / |
| /---|--------- boy
| / x|
|/_____|
70m
In this triangle, the height of the tower is represented by 'h' and the distance between the boy and the flag is 70m. The angle of elevation from the boy's perspective for the top of the tower is 70 degrees, while the angle of elevation for the bottom of the tower is 69 degrees.
Using trigonometry, we know that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side. Therefore, we can set up the following equation:
tan(70 degrees) = h / 70m
tan(69 degrees) = (h + x) / 70m
We need to find the value of 'h,' the height of the tower. To eliminate 'x' from the equation, we can subtract the second equation from the first equation:
tan(70 degrees) - tan(69 degrees) = h / 70m - (h + x) / 70m
Now, we can simplify the equation:
[tan(70 degrees) - tan(69 degrees)] * 70m = (h - h - x) / 70m
[tan(70 degrees) - tan(69 degrees)] * 70m = x / 70m
By solving for 'x,' we can find the distance between the bottom of the tower and the boy.
x = [tan(70 degrees) - tan(69 degrees)] * 70m
Once we have the value of 'x,' we can substitute it back into either equation to calculate the height of the tower 'h.'
h = tan(70 degrees) * 70m
Therefore, by substituting the values of tan(70 degrees) and tan(69 degrees) and performing the calculations, we can find the value of 'h' and determine the height of the tower.