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

[mugzieee]

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...

 

[chroot]

It's just a proportion. 360 degrees around the equator is about 25,000 miles.

$$\frac{360^o}{25,000\, \textrm{mi}} = \frac{x}{5000\, \textrm{mi}}$$

Solve for x.

- Warren

 

[mugzieee]

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?

 

[Data]

25000 miles is the circumference of the Earth at the equator (approximately). In your question, it would be better to use $$2 \pi (4000 \mbox{miles} )$$, though, since the question gives you the radius of the Earth as 4000 miles.

 

[mugzieee]

thanks for the input guys. but, if we wanted to solve this problem trigonometrically, what would be a good way to do it?

 

[chroot]

It's a circle, man. There's no trigonometry involved. You can use the arc-length formula if you'd like: $s = r\theta$, which is essentially what I already did.

- Warren