Since the angle formed by the tangent satellite signals is 104°, the angle between the tangent and the radius connecting the Earth's center to the point of intersection is 180° - 104° = 76°.
Let's call the length of the tangent line segment "a" and the radius of the Earth "r". The line segment from the center of the Earth to the point of intersection forms a right angle triangle with the tangent line segment, so we can use trigonometry to find the length of this line segment.
Using the tangent function, we get:
tan(76°) = opposite / adjacent
tan(76°) = x / r
x = r * tan(76°)
Now, we know that the length of the entire circumference of the Earth is 2πr, so the fraction of the circumference intercepted by the tangent line segment is:
x / (2πr)
Substituting in x = r * tan(76°), we get:
r * tan(76°) / (2πr)
Canceling out the r terms, we get:
tan(76°) / (2π)
Using a calculator to evaluate this expression, we get:
0.023148
To find the angle of the intercepted arc on Earth, we need to multiply this fraction by 360°:
0.023148 * 360° = 8.3448°
But we're looking for the supplement of this angle (the angle formed by the major arc, not the minor arc), so we subtract this from 360°:
360° - 8.3448° = 351.6552°
Finally, we substitute back into the theorem mentioned in the hint:
360° - x = 351.6552°
Solving for x:
x = 360° - 351.6552°
x = 8.3448°
Therefore, the measure of the intercepted arc on Earth is approximately 8.3448°, and the answer should be rounded to the nearest whole degree, which is 8°.
The farthest distance a satellite signal can directly reach is the length of the tangent to the curve of Earth's surface. If the angle formed by the tangent satellite signals is 104°, what is the measure of the intercepted arc (x) on Earth?
Hint: If minor arc AC = x then remember that major arc ABC = 360 - x. You will need to substitute into your equation provided in the theorem.
Show all work.
(This is 10th grade math, please don't make it advanced, shorten the steps.)
(You should get 76°.)
1 answer