Angle of Turn from Kampala to Singapore: Solving with Law of Cosines

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In summary, the Earth's radius is 4000 miles and Kampala and Singapore are both located near the equator, with a distance of 5000 miles between them. To determine the angle traveled between the two cities relative to the Earth, a proportion can be set up using the circumference of the Earth at the equator (approximately 25,000 miles) and the distance between the two cities. This can also be solved using the arc-length formula (s = r\theta) since the distance and radius are given.
  • #1
mugzieee
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The Earth's radius is about 4000 miles. Kampala, the capital of Uganda, and Singapore are both nearly on the equator. The distance between them is 5000 miles.Through what angle do you turn, relative to the earth, if you fly from Kampala to Singapore?

the only thing i can think of doing is using the law of cosines, if both sides equal 4000 miles and the other side of the tirangle is equal to 5000k miles...but i tried that it doesn't work..i tried using all the trig functions and it still didnt work..would someone just point me to the direction that will get me started...
 
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  • #2
It's just a proportion. 360 degrees around the equator is about 25,000 miles.

[tex]\frac{360^o}{25,000\, \textrm{mi}} = \frac{x}{5000\, \textrm{mi}}[/tex]

Solve for x.

- Warren
 
  • #3
how did you know how to set up that proportion, I am sorry i don't see it too clearly. i mean your method was correct kuz i got the correct answer, but i just don't see how u set up the proportion. how did you know that 360degrees around the Earth is 25000 miles?
 
  • #4
25000 miles is the circumference of the Earth at the equator (approximately). In your question, it would be better to use [tex] 2 \pi (4000 \mbox{miles} )[/tex], though, since the question gives you the radius of the Earth as 4000 miles.
 
  • #5
thanks for the input guys. but, if we wanted to solve this problem trigonometrically, what would be a good way to do it?
 
  • #6
It's a circle, man. There's no trigonometry involved. You can use the arc-length formula if you'd like: [itex]s = r\theta[/itex], which is essentially what I already did.

- Warren
 

1. What is the "Angle of Turn" in this problem?

The "Angle of Turn" refers to the change in direction from Kampala to Singapore. It is the angle formed between the initial direction (from Kampala) and the final direction (towards Singapore).

2. How is the Law of Cosines used to solve this problem?

The Law of Cosines is used to solve this problem by calculating the angle of turn using the sides of a triangle. The formula for the Law of Cosines is c² = a² + b² - 2ab cos(C), where c is the side opposite the angle of turn, and a and b are the other two sides of the triangle. By plugging in the values for the sides of the triangle, the angle of turn can be calculated using the inverse cosine function.

3. What information is needed to solve this problem?

To solve this problem, the distance between Kampala and Singapore and the lengths of the sides of the triangle formed by the two points and the Earth's center are needed. The distance between the two points can be found using a map or online tool, and the lengths of the sides can be calculated using their respective coordinates and the Earth's radius.

4. Can the Law of Cosines be used for any angle of turn problem?

Yes, the Law of Cosines can be used for any angle of turn problem as long as the necessary information is known. It is a general formula for solving triangles and can be applied to any case, including this problem of finding the angle of turn from Kampala to Singapore.

5. Are there any limitations to using the Law of Cosines for this problem?

One limitation to using the Law of Cosines for this problem is that it assumes a perfect spherical shape for the Earth. However, the Earth's shape is not a perfect sphere, and this may cause slight discrepancies in the calculated angle of turn. Additionally, the Law of Cosines may not be efficient for solving this problem if the distance between the two points is very large, as it involves a more complex calculation compared to other methods.

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