From an observation tower that overlooks a runway, the angles of depression of point A, on one side of the runway, and point B, on the opposite side of the runway are 6 degrees and 13 degrees, respectively. The points and the tower are in the same vertical plane and the distance from A to B is 1.1 km. Determine the height of the tower.
Please verify if 72.2 meters is the height of the tower.
Label the figure, T is the top of the tower, Dis the base of the tower.
angle DAT=13
angle DB=6
AB = 1100m
find DT
from the figure, AT/sin13=1100/Sin(13-6) solve for AT.
then, AT/sin90=DT/sin13
Thanks for your help, I guess we took different approaches to this so now I'm a little confused. Is it possible to explain a little clearer?
I drew the figure. I saw two triangles. In one, I used the law of sines (I knew two angles, one side) to find the other side, then used that side (common to the other triangle, to solve height with the law of sines again.
Wonderful help. Thank you.
To determine if 72.2 meters is the height of the tower, we can use trigonometric ratios and the given information. Let's begin by drawing a diagram to visualize the situation.
From the top view of the observation tower, we have point A and point B on either side of the runway. The angles of depression from the observation tower to points A and B are 6 degrees and 13 degrees, respectively. We need to find the height of the tower.
Let's label the height of the tower as "h" and add it to our diagram.
Now, we can use the tangent function to relate the angles of depression to the height of the tower.
Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle.
For point A:
tan(6 degrees) = opposite side / adjacent side
tan(6 degrees) = h / x
x = h / tan(6 degrees) ...(1)
For point B:
tan(13 degrees) = opposite side / adjacent side
tan(13 degrees) = h / y
y = h / tan(13 degrees) ...(2)
Given that the distance between points A and B is 1.1 km, we have:
x + y = 1.1 km
Substituting the values from equations (1) and (2) into the equation x + y = 1.1 km, we can eliminate the variables x and y:
h / tan(6 degrees) + h / tan(13 degrees) = 1.1 km ...(3)
Now, we can solve equation (3) to find the value of h.
Using the given values of the angles of depression, we have:
h / tan(6 degrees) + h / tan(13 degrees) = 1.1 km
h / tan(6 degrees) + h / tan(13 degrees) = 1.1 * 1000 meters (converting km to meters)
Substitute the values for the angles of depression:
h / tan(6 degrees) + h / tan(13 degrees) = 1.1 * 1000 meters
h / 0.1051 + h / 0.2249 = 1100
Now, we can solve this equation to find the value of h, the height of the tower:
Multiply through by the least common denominator (0.1051 * 0.2249) to clear fractions:
0.2249 * h + 0.1051 * h = 1100 * 0.1051 * 0.2249
Combine like terms:
0.3300 * h = 256.9798
Finally, divide both sides by 0.3300:
h = 256.9798 / 0.3300
h ≈ 778.725 meters
Therefore, the height of the tower is approximately 778.725 meters. The value of 72.2 meters does not match the calculated height, so it is incorrect.