anonymous
  • anonymous
In a circle of radius 7 miles, find the length of the arc that subtends a central angle of 5 radians. Where would i start on this?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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phi
  • phi
you could set up a ratio circumference (all the way round) is to 2pi radians as sector is to 5 radians
phi
  • phi
2 pi radians is about 6.3 radians 5 radians will be most of the way round
phi
  • phi
or you can remember s= r \( \theta\) where s is arc length, r is radius, and θ is the central angle in radians

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anonymous
  • anonymous
what exactly is the central angle? i did the problem and got like a crazy number, and it said the answer was 35.
phi
  • phi
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anonymous
  • anonymous
i get it now actually i was putting 5 radians as 5pi before and thats why it was so large
phi
  • phi
the formula is easy to use: radius times angle (but the angle has to be in radians, not degrees) = arc length. 7*5= 35 the ratio idea is that all the way around the circle C= 2 pi r \[ \frac{2 \pi r}{2 \pi} = \frac{s}{\theta} \]
anonymous
  • anonymous
before i was thinking that radians always had to have pi involved in them but i realized that they have pi to make them easier to look at and use
phi
  • phi
yes, people attach pi to radians so much you think it is part of the definition. but radians are really numbers from 0 to 6.3 (roughly). The reason we use pi so much (as in pi/3 radians) is you get nice whole numbers when you convert to degrees. In simple problems you don't want to see 3 radians ... because that will be ugly if you convert it to degrees)
anonymous
  • anonymous
for the problem, Find the distance along an arc on the surface of earth that subtends a central angle of 5 minutes (1 minute= 1/60th of degree). The radius of the earth is 3960 miles
anonymous
  • anonymous
would i use the same formula? or a velocity one? thanks for the help that clears things up
phi
  • phi
velocity ? no. this is asking for the length of an arc the easiest thing is change the minutes to degrees and then to radians then use s = r * theta
anonymous
  • anonymous
okay, thanks a lot
phi
  • phi
you should get 5 nautical miles (which are slightly bigger than statute miles)
anonymous
  • anonymous
i came out to 5.795, is that correct?
phi
  • phi
yes see http://www.wolframalpha.com/input/?i=5.795+miles+to+nautical+miles notice you get very close to 5 nautical miles. 1 minute of arc corresponds to 1 nautical mile. Navigators use nautical miles because they can convert to degrees and minutes of latitude (and longitude) easily
anonymous
  • anonymous
okay i see thank you. my next problem says that 3=(theta)6. i come up with .5 radians, how would i convert this. would i mulitply its by 180/pi?
phi
  • phi
yes
phi
  • phi
or leave it 0.5 radians
anonymous
  • anonymous
okay cool. on the area of a sector formula is it \[a=(1/2)(\theta)(r^2)? or 1/2(\theta r^2)?\]
phi
  • phi
I never memorized a sector formula for area. I figure what fraction of a circle's area ? pi r^2 * theta/2pi which simplifies to (1/2) theta * r^2
anonymous
  • anonymous
okay, i was just wondering if it was to be mulitplyied by 1/2 at the end or something
anonymous
  • anonymous
does it not matter the order in which you multiply items in math?

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