An observer at A looks due north and sees a meteor with an angle of elevation of 70deg. At the same instant, another observer 30miles east of A, sees the same meteor and apprroximates its position as N 50deg W but fails to note its angle of elevation. Find the height of the meteor and its distance form A.
The line of sight of the 2nd observer
is represented by the hypotenuse of a rt triangle, and the height of the
meteor is represented by the ver. side
of the rt triangle.The acute angle bet.
ween the ver. side and hyp. = 50 deg.
The angle of elevation is equal to the
other acute angle or 40 deg.
Tan(40) = h/30,
h =30*Tan(40) = 25.2 =height of
meteor
Tan(70) = 25.2 / d,
d = 25.2/Tan(70) = 9.2 = distance of
meteor.
CORRECTION!!
Sin(70) = 25.2/d,
d = 25.2/Sin(70) =26.8.
To solve this problem, we can use the concept of triangulation, which involves creating a triangle using the observers and the meteor.
Let's begin by drawing a diagram. Place observer A at the bottom of the page and observer B to the right of A. Connect A and B with a straight line to represent the distance between them. Now, draw a line upwards from A to represent the meteor and an angle of 70 degrees with respect to the observer's line of sight.
Next, draw a line from B towards the meteor, but at an angle of 50 degrees west of north. Since the observer at B does not note the angle of elevation, we assume that the meteor is somewhere on this line.
Now, we have a triangle with observer A, observer B, and the meteor. To find the height of the meteor and its distance from observer A, we need to calculate the lengths of the sides of this triangle.
Since we have an angle of elevation of 70 degrees at observer A, we can determine that the angle between the observer's line of sight and the ground is 90 - 70 = 20 degrees. This means that the triangle formed by observer A, the meteor, and the ground is a right triangle.
Now, let's calculate the height of the meteor using the tangent function. Tangent is defined as opposite/adjacent, so:
tangent(20 degrees) = height of the meteor / distance between A and the meteor
We know that the distance between observer A and observer B is 30 miles. However, we don't have the distance between observer B and the meteor. To find this distance, we can use the sine function.
sine(50 degrees) = distance between B and the meteor / distance between A and the meteor
Now we have two equations:
tangent(20 degrees) = height of the meteor / 30 miles
sine(50 degrees) = distance between B and the meteor / 30 miles
We can rearrange the first equation to solve for the height of the meteor:
height of the meteor = tangent(20 degrees) * 30 miles
Using a scientific calculator, calculate tangent(20 degrees) ≈ 0.364
height of the meteor ≈ 0.364 * 30 miles ≈ 10.92 miles
Now, we can rearrange the second equation to solve for the distance between B and the meteor:
distance between B and the meteor = sine(50 degrees) * 30 miles
Using a scientific calculator, calculate sine(50 degrees) ≈ 0.766
distance between B and the meteor ≈ 0.766 * 30 miles ≈ 22.98 miles
Therefore, the height of the meteor is approximately 10.92 miles, and the distance between observer A and the meteor is approximately 22.98 miles.