The farthest distance a satellite signal can directly reach is the length of the segment tangent to the curve of Earth's surface. If the angle formed by the tangent satellite signals is 138°, what is the measure of the intercepted arc on Earth?
We begin by drawing a diagram:
Let O be the center of the Earth, and let A and B be two points on the curve of the Earth's surface that are tangent to the satellite signal. Let C be the midpoint of segment AB, and let D be the point of intersection of the segment OC with the Earth's surface. We want to find the length of arc AC.
Since AD is perpendicular to segment AB, triangle ACD is a right triangle. Let x be the length of AC, and let r be the radius of the Earth. Then we have:
AC = 2r sin(69°)
AD = r cos(69°)
CD = r sin(69°)
By the Pythagorean theorem, we have:
(2r sin(69°))^2 = (r + r cos(69°))^2 + (r sin(69°))^2
Simplifying this equation, we get:
4r^2 sin^2(69°) = r^2 + 2r^2 cos(69°) + r^2 sin^2(69°)
Solving for x, we get:
x = 2r sin(69°) = r(2 sin(69°))/(1 + 2cos(69°)) ≈ 3086 km
Therefore, the measure of arc AC is approximately 3086 km.
To find the measure of the intercepted arc on Earth, we need to find the central angle associated with the tangent segment.
The central angle is equal to the angle formed by the tangent satellite signals, which is given as 138°. We know that the tangent segment forms a right triangle with the radius of the Earth.
In a right triangle, the central angle is equal to the angle formed by the tangent line (138°) plus the angle formed by the radius (90°).
Central angle = 138° + 90°
Central angle = 228°
Now, we can use the formula for calculating the measure of an intercepted arc:
Intercepted arc = (central angle / 360°) x circumference of the Earth
The circumference of the Earth is approximately 40,075 km.
Intercepted arc = (228° / 360°) x 40,075 km
Intercepted arc ≈ (0.6333) x 40,075 km
Intercepted arc ≈ 25,398.77 km
Therefore, the measure of the intercepted arc on Earth is approximately 25,398.77 km.