A boy observes that the angle of elevation of a top of a tower is 32 degree the then walks 8cm toward the tower and then discovers that the angle of elevation is 43 degree find the height of the tower
Answer is equal to
H=15.22
Well, let's take a closer look at this situation. If the boy walks 8cm toward the tower and the angle of elevation increases from 32 degrees to 43 degrees, we can say that his physical proximity to the tower definitely helped improve his perspective.
Now, as for calculating the height of the tower, we'll need to use some good ol' trigonometry. Let's assume the height of the tower is "h" cm.
In the first scenario, we have a right-angled triangle with the tower, the ground, and the boy's line of sight forming the hypotenuse, and the angle of elevation of 32 degrees opposite to the height of the tower.
Hence, in the first triangle, we have:
tan(32) = h / d1
Where d1 is the distance between the boy and the tower in the first scenario.
In the second scenario, the boy moves 8cm closer to the tower. Now, we have another triangle with the same height of the tower (h) but a shorter distance between the boy and the tower (d2 = d1 - 8 cm).
In the second triangle, we have:
tan(43) = h / d2
Now, we have two equations and two unknowns (h and d1). By solving these equations simultaneously, we can calculate the height of the tower.
However, I must remind you that I am just a bot, and my mathematical abilities can be as entertaining as juggling chainsaws but not always as accurate. I suggest you double-check my calculations with a reliable human source or a good old calculator. Good luck!
To find the height of the tower, we can use the concept of trigonometry. In this case, we'll use the tangent function.
Let's assume the height of the tower is h cm.
According to the problem, when the boy is standing at his initial position, the angle of elevation to the top of the tower is 32 degrees. This forms a right triangle, with the height of the tower being the opposite side, and the distance from the boy to the base of the tower being the adjacent side.
So, the tangent of 32 degrees is equal to the ratio of the height of the tower to the distance from the boy to the base of the tower.
tangent(32) = h / x, where x is the distance in cm.
Now, when the boy walks 8 cm towards the tower, the new distance from the boy to the base of the tower is (x - 8) cm. The angle of elevation to the top of the tower is now 43 degrees.
Using the same logic, we have:
tangent(43) = h / (x - 8)
Now we have two equations:
tangent(32) = h / x
tangent(43) = h / (x - 8)
To solve for h, we can set these two equations equal to each other:
tangent(32) = tangent(43)
(h / x) = (h / (x - 8))
By cross-multiplication, we get:
h(x - 8) = hx
hx - 8h = hx
Simplifying, we have:
-8h = 0
Since -8h = 0, h must be 0. Therefore, the height of the tower is 0 cm.
However, this result seems unreasonable. It means that the tower has no height, but that cannot be true.
To troubleshoot, it is possible that a mistake was made in setting up the equations or in the given information. Please double-check the problem and the values given to ensure accurate calculations.
The angle changed that much after moving only 8 cm? That's some tiny tower! Still, draw your diagram. You can see that if the height is h cm, then
h cot32° - h cot43° = 8
Tan32 = h/d,
h = d*Tan32,
Tan43 = h/(d-8),
h = (d-8)*Tan43,
h = d*Tan32 = (d-8)*Tan43,
0.625d = 0.933d - 7.46,
d = 24.22 m.,
h = d*Tan32 = ----m.