From an observation tower that overlooks a runway, the angles of depressin of point A, on one side of the runway, and point B, on the opposite side of the runway are, 6degree and 13degree, respectively. The points and the tower are in the same vertical plance and the distance from A and B is 1.1km. Determine the height of the tower.

Question

From an observation tower that overlooks a runway, the angles of depressin of point A, on one side of the runway, and point B, on the opposite side of the runway are, 6degree and 13degree, respectively. The points and the tower are in the same vertical plance and the distance from A and B is 1.1km. Determine the height of the tower.

Answer 1

Draw 2 rt triangles witha common ver. side.

h = Y = Ver. side.
1.1km+X = Hor. side.
X = dist. from hyp. of smaller triangle to base of tower.

tan6 = h/(1.1+X).
h = (1.1+X)tan6.

tan13 = h/X
h = X*tan13.

h = (1.1+X)tan6 = X*tan13
(1.1+X)tan6 = X*tan13
1.1*tan6+X*tan6 = X*tan13
X*tan6-X*tan13 = -1.1*tan6
0.1051x-0.2309x = -0.1156
-0.12577x = -0.1156
X = 0.92 km.

h = X*tan13 = 0.212 km. = 212 m.

Answer 2

To determine the height of the tower, we can use the concept of trigonometry and the angles of depression.

Let's denote the height of the tower as 'h'. The distance from point A to the tower can be represented as 'x', and the distance from point B to the tower can be represented as 'y'.

From the given information, we know that the points A and B, along with the tower, are in the same vertical plane, so we have a right-angled triangle formed by the tower, point A, and point B.

We can use the tangent function to relate the angles of depression to the distances and the height of the tower.

For point A:
tan(6°) = h / x

For point B:
tan(13°) = h / y

We also know that the distance between points A and B is 1.1 km, so x + y = 1.1.

Now, we have two equations with two variables, x and y:
tan(6°) = h / x
tan(13°) = h / y
x + y = 1.1

To solve this system of equations, we can use substitution or elimination method. In this case, let's use the substitution method.

From the first equation, we can express x in terms of h:
x = h / tan(6°)

Substituting this value of x in the third equation:
(h / tan(6°)) + y = 1.1

Rearranging this equation, we get:
y = 1.1 - (h / tan(6°))

Now, we substitute the values of x and y in the second equation:
tan(13°) = h / (1.1 - (h / tan(6°)))

Simplifying this equation will give us the value of h, the height of the tower.

Using a scientific calculator or an online trigonometry calculator, we can calculate the value of h.