À boy observés that the angle of élévation of the top of the tower is 32degree and the boy moves 8m to the tower and sees the angle of élévation is 43 find the height of the tower
The traditional way to do this style of question has you ending up with
height = 8*tan32*tan43/(tan43 - tan32) = ....
I found through the years that students follow more readily the following method:
Label the top of the tower P and its bottom Q
label the boy's original position A and his new position B
It is easy find all the angles in triangle ABP.
then in triangle ABP:
BP/sin32° = 8/sin11°
BP = 8sin32/sin11
in the right-angled triangle BPQ
sin 43° = h/BP
h = BPsin43 = 8sin32sin43/sin11 = .... for the same answer
or, as a slightly less opaque version of the first solution, if the height is h, then
h cot32° - h cot43° = 8
But you do have to be comfortable thinking in terms of cotangent, rather than tangent.
X = hor. distance to base of tower.
x-8 = 8 m close.
Tan32 = h/x.
h = x*Tan32.
Tan43 = h/(x-8)
h = (x-8)*Tan43.
h = x*Tan32 = (x-8)*Tan43
0.62x = (x-8)0.93
X = 24 m.
h = x*Tan32 = 24*0.62 = 15 m.
To find the height of the tower, we'll use the concept of trigonometry. Let's assume the height of the tower is 'h' meters.
Step 1: Draw a diagram representing the scenario. Label the height of the tower as 'h' and the distance the boy moves towards the tower as 'x'.
Step 2: Identify the angles involved. The first angle of elevation is 32 degrees, and the second angle of elevation is 43 degrees.
Step 3: Consider the right-angled triangle formed by the boy, the distance he moved (x), and the height of the tower (h). From trigonometry, the tangent function relates the opposite and adjacent sides of a right triangle.
In this case, the tangent of the first angle of elevation (32 degrees) is equal to the opposite side (h) divided by the adjacent side (x).
Therefore, tan(32°) = h / x ----------- Equation 1
Step 4: Now consider the triangle formed by the boy after he moved towards the tower. The base of this triangle is (x + 8) meters.
From trigonometry, the tangent of the second angle of elevation (43 degrees) is equal to the opposite side (h) divided by the adjacent side (x + 8).
Therefore, tan(43°) = h / (x + 8) ----------- Equation 2
Step 5: Now we have a system of two equations (Equations 1 and 2) with two unknowns (h and x). We can solve these equations simultaneously to find the value of 'h'.
Rearrange Equation 1 to solve for h:
h = x * tan(32°)
Replace h in Equation 2:
tan(43°) = (x * tan(32°)) / (x + 8)
Step 6: Solving the equation above will give us the value of 'x'. Substitute this value in Equation 1 to find the value of 'h'.
After calculating the values, the height of the tower can be determined.