The angles of elevation of the top of a tower as observed from the bottom and top of a building of height 6th are 60° and 45° respectively.The distance of the base of the tower from the base of the building?
If the tower has height h and is at a distance x, then
h/x = tan 60°
(h-6th)/x = tan 45°
Eliminating h, we have
x/tan 60° = x tan 45° + 6th
x(1-1/√3) = 6th
x = 6th/(1-1/√3)
whatever 6th is supposed to mean.
To find the distance of the base of the tower from the base of the building, we can use the concept of trigonometry and the given angles of elevation.
Let's denote the distance from the base of the building to the base of the tower as "x".
First, let's consider the angle of elevation of 60 degrees. This means that the angle formed between the line of sight from the bottom of the building to the top of the tower and the horizontal ground is 60 degrees.
Using trigonometry, we can determine the height of the tower. The height of the tower will be the opposite side of the triangle formed, and the distance from the base of the building to the base of the tower will be the adjacent side. Since we know the angle and the height, we can use the tangent function to find x:
tan(60°) = height of the tower / x
tan(60°) = h / x .....(1)
Similarly, for the angle of elevation of 45 degrees, the height of the tower will be the opposite side and the distance from the top of the building to the top of the tower will be the adjacent side. We can again use the tangent function to find the height:
tan(45°) = height of the tower / 6
tan(45°) = h / 6 .....(2)
Now we have two equations (equations 1 and 2) with two unknowns (h and x). We can solve these equations simultaneously to find their values.
From equation (1), we have:
tan(60°) = h / x
√3 = h / x (since tan(60°) = √3)
h = √3x .....(3)
We can substitute this value of h in equation (2):
tan(45°) = h / 6
1 = √3x / 6 (since tan(45°) = 1)
√3x = 6
x = 6 / √3
To simplify the answer, we can rationalize the denominator by multiplying both the numerator and the denominator by √3:
x = (6 / √3) * (√3 / √3)
x = (6√3) / 3
x = 2√3
Therefore, the distance from the base of the tower to the base of the building is 2√3 (approximately 3.46) units.