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°.)
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°.
To solve this problem, we can use the information provided and set up an equation using the tangent line and the intercepted arc. Let's follow these steps:
Step 1: Recall the theorem mentioned in the hint, which states that if minor arc AC = x, then major arc ABC = 360 - x.
Step 2: Given that the angle formed by the tangent satellite signals is 104°, we can let x represent the intercepted arc.
Step 3: According to the theorem, the major arc CBC (the whole circle) is equal to 360 - x. Since the whole circle is 360°, we can set up an equation:
360 - x = 360 - 104
Step 4: Simplify the equation:
-x = -104
Step 5: Multiply both sides of the equation by -1 to isolate x:
x = 104
So, the measure of the intercepted arc is 104°. However, this contradicts the provided answer of 76°. Please check the given information and verify the accuracy of the answer.