The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations A and B, which are 95 mi apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 87.0° and 84.2°, respectively. (Round your answers to the nearest mile.)
(a) How far is the satellite from station A?
_____mi
(b) How high is the satellite above the ground?
_____mi
To find the distance from station A to the satellite, we can use trigonometry. Let's assume that the satellite is at point S directly above the line segment connecting A and B, and let x represent the distance from A to the satellite.
We can create a right triangle with the following information:
- The angle of elevation at A is 87.0°. This means that the angle between the line segment AB and the line segment AS is also 87.0°.
- The angle of elevation at B is 84.2°. This means that the angle between the line segment AB and the line segment BS is also 84.2°.
- The length of AB is given as 95 miles.
Now, let's use the tangent function to find the value of x:
tan(87.0°) = (x / 95)
x = 95 * tan(87.0°)
x ≈ 394.18 miles
Therefore, the distance from station A to the satellite is approximately 394 miles.
To find the height of the satellite above the ground, we can use the angle of elevation at A. The side opposite this angle is the height of the satellite, and the side adjacent to it is the distance from station A to the satellite (x).
Using trigonometry, we can use the tangent function again:
tan(87.0°) = (height / x)
height = x * tan(87.0°)
height ≈ 394 * tan(87.0°)
height ≈ 720.78 miles
Therefore, the height of the satellite above the ground is approximately 721 miles.
Draw a diagram. If the height is h,
h cot84° - h cot87° = 95
(a) the distance d is found via
h^2+(h cot87°)^2 = d^2