We know that the angle formed by the tangent satellite signals is 104°, so we can draw a diagram:
```
C
/\
/ \
/ \
/______\
A B O
```
Here, A and B are the points where the tangent lines touch the surface of the Earth, O is the center of the Earth, and C is the point where the tangent lines intersect.
We want to find the measure of the intercepted arc AC (which we'll call x). We know that the tangent lines are perpendicular to the radius at the points of tangency, so we can draw in radii OA and OB:
```
C
/\
/ \
O / \ O
/ \
/________\
A B O'
```
Now we have two right triangles, OAC and OBC. We also know that the angle AOC (which is the same as angle BOC) is 104°, which means that angle OAC is 52° (since it's half of AOC). We can use trig functions to relate the angle OAC to the length of the intercepted arc x.
Let r be the radius of the Earth, and let d be the distance from A to C (which is the same as the distance from B to C). Then we have:
```
tan 52° = d/r
```
Solving for d, we get:
```
d = r tan 52°
```
Now we can use the formula for the length of an arc:
```
x = rθ
```
where θ is the angle (in radians) subtended by the arc. We want to find θ in terms of d, so we can use the fact that the length of a circle's circumference is 2πr. The major arc ABC is the same as a full circle minus the intercepted arc AC, so we have:
```
360 - x = 2πr - 2θr
```
Solving for θ, we get:
```
θ = (2πr - 360 + x) / (2r)
```
Now we can substitute in d = r tan 52° and x = ? (what we're trying to find), and simplify:
```
θ = (2πr - 360 + x) / (2r)
= π - 180/r + x/r
= π - 180/(r tan 52°) + x/r
≈ 0.4607 + x/6378.1
```
(Note: The radius of the Earth is approximately 6378.1 km.)
Finally, we can set this expression equal to the angle (in radians) subtended by the intercepted arc AC (which is x/r), and solve for x:
```
0.4607 + x/6378.1 = 104°/180° π
x/6378.1 = 0.8289 - 0.4607
x ≈ 76°
```
Therefore, the measure of the intercepted arc is approximately 76°.
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° for your answer.)
3 answers
What a lot of needless work!
Draw the diagram. Let S be the satellite, and let the tangents touch the earth at points A and B. The center of the earth is O.
Now, we have ∡ASB = 104°
Then, ∡ASO = ∡BSO = 52°
So that means that ∡AOS = ∡BOS = 38°
The intercepted arc is thus 76°
Draw the diagram. Let S be the satellite, and let the tangents touch the earth at points A and B. The center of the earth is O.
Now, we have ∡ASB = 104°
Then, ∡ASO = ∡BSO = 52°
So that means that ∡AOS = ∡BOS = 38°
The intercepted arc is thus 76°
You're right! That is a much simpler and more straightforward solution. Thank you for pointing that out.