Two observers 1 km apart spot an airplane. The angles of elevation of the airplane from the two observers are 22 degrees 50 minutes and 24 degrees 40 minutes. If the observers and the airplane are in the some vertical plnae, compute the altitude of the airplane.
ok i don\'t know how to do this if you could show me how to do this
Also I think i may be reading this wrong i don\'t know
they put 22 then raise it to a circle which is used for degrees but than after that they do something like 50\' which i believe are minutes
a2 = unkown horizontal distance from observer to verticle component that the line of sight makes at the air plane
a2 tan 24.7 = opposite
(1 km + a2) tan 22.8 = x
a2 tan 24.7 = (1 km + a2) tan 22.8
not sure how to solve for a2
let's make a diagram.
(I think your's is probably close)
draw a straight line and mark two points A and B and label AB = 1 km
Place the plane above the line AB and to the right of it, call it P
Extend AB, and from P draw a perpendicular down to the extended line AB and let Q be the point where it meets AB extended.
You should now have two right-angled triangles, APQ and BPQ, with the right angle at Q.
Label BQ as x, and PQ as y
now in the inside triangle,
tan 24.7º = y/x
y = xtan24.7
in the larger triangle,
tan 22.8 = y/(x+1)
y = (x+1)tan22.8
so (x+1)tan22.8 = xtan24.7
expanding
xtan22.8 + tan22.8 = xtan24.7
tan22.8 = xtan24.7 - xtan22.8
tan22.8 = x(tan24.7 - tan22.8)
x = tan22.8/(tan24.7-tan22.8)
now use your calculator to find x (I don't mine handy)
once you have x, sub it back into
y = xtan24.7 to find the height y
0.2
To solve for the unknown altitude (a2) of the airplane, we can use the concept of trigonometry and set up some equations based on the given information.
Let's define the given variables:
- d: distance between the two observers (1 km)
- x: horizontal distance from one observer to the vertical component that the line of sight makes at the airplane
- a2: altitude of the airplane (what we want to find)
- θ1: angle of elevation from the first observer (22 degrees 50 minutes)
- θ2: angle of elevation from the second observer (24 degrees 40 minutes)
First, let's convert the angle measures from degrees and minutes to decimal degrees. One degree is equal to 60 minutes. So, we have:
θ1 = 22 + 50/60 = 22.8333 degrees
θ2 = 24 + 40/60 = 24.6667 degrees
Now, we can set up the following equations:
Equation 1:
a2 * tan(θ2) = x
Equation 2:
(1 km - x) * tan(θ1) = x
The logic behind Equation 1 is that the tangent of the angle of elevation (θ2) gives us the ratio of the altitude (a2) to the horizontal distance (x).
In Equation 2, we need to consider the fact that the horizontal distance from the second observer to the vertical component of the line of sight is (1 km - x), and then we multiply it by the tangent of the angle of elevation (θ1) to get the ratio of the altitude (a2) to the horizontal distance (x).
To solve these equations, we can substitute the value of x from Equation 1 into Equation 2 and solve for a2. Here are the steps:
1. Substitute x in Equation 2:
(1 km - x) * tan(θ1) = a2 * tan(θ2)
2. Distribute and rearrange the equation:
tan(θ1) - x * tan(θ1) = a2 * tan(θ2)
x * tan(θ1) + a2 * tan(θ2) = tan(θ1)
3. Solve for a2:
a2 = tan(θ1) / tan(θ2) - x * tan(θ1) / tan(θ2)
Now, you can substitute the values of θ1, θ2, and x, and calculate a2 using a scientific calculator or trigonometric tables.
Note: Make sure to use your calculator in degree mode when calculating the tangents.