the angle of elevation of the top of tower from a point A due south of it is 68 degrees .From a point B due east of A the angle of elevation of the top is 54 degrees.If A is 50 m from B find the height of the tower.
18.73 degree
To find the height of the tower, we can use the concept of trigonometry. Let's visualize the problem first:
We have a tower, a point A due south of the tower, and a point B due east of A. We are given that the angle of elevation of the top of the tower from point A is 68 degrees, and from point B is 54 degrees. The distance between points A and B is given as 50 meters.
Let's label the height of the tower as h.
From point A, we can form a right triangle with the base being the distance between A and the tower, and the height being the height of the tower itself. The angle between the base and the height is the angle of elevation from point A.
Using trigonometry, we can say that:
tan(68 degrees) = h / x... (equation 1)
where x is the distance between A and the tower.
Similarly, from point B, we can form another right triangle with the base being the distance between B and the tower, and the height being the height of the tower itself. The angle between the base and the height is the angle of elevation from point B.
Using trigonometry, we can say that:
tan(54 degrees) = h / (x + 50)... (equation 2)
We now have a system of two equations. Let's solve them simultaneously to find the value of h.
Step 1: Solve Equation 1 for x:
x = h / tan(68 degrees)
Step 2: Substitute the value of x in Equation 2:
tan(54 degrees) = h / (h / tan(68 degrees) + 50)
Simplifying this equation will give us the value of h, which is the height of the tower.
To find the height of the tower, we can make use of the trigonometric functions.
Let's denote the height of the tower as 'h'.
First, let's draw a diagram to help visualize the problem.
A
/ |
/ | h
68° / |
/ |
/ |
/θ θ | B
/________|
Now, let's consider the triangle formed by the tower's height, distance from A to the tower, and the angle of elevation (θ) from point A:
/|
/h|
/ |
68° / θ |
/ |
/_____|
Using the tan function, we get:
tan(θ) = h / distance from A to the tower
tan(68°) = h / distance from A to the tower
Now, let's consider the triangle formed by the distance from A to B, the height of the tower, and the angle of elevation (θ) from point B:
B
|\
| \h
| \
θ | \
| \
|____\
Using the tan function, we have:
tan(θ) = h / distance from A to B
tan(54°) = h / 50m
Now, we have two equations:
tan(68°) = h / distance from A to the tower -- (1)
tan(54°) = h / 50m -- (2)
Simplify equation (1):
h = (tan(68°)) * distance from A to the tower
Substitute the value of h from equation (1) into equation (2):
(tan(68°)) * distance from A to the tower = (tan(54°)) * 50m
Now, we can solve for the height (h) of the tower. Let's assume the distance from A to the tower is 'x':
(tan(68°)) * x = (tan(54°)) * 50m
Solve for x:
x = ((tan(54°)) * 50m) / (tan(68°))
Now, substitute this value of x back into equation (1) to find the height of the tower (h):
h = (tan(68°)) * x
Finally, calculate the height of the tower using the given values.