anonymous
  • anonymous
find the length of the arc on a circle with a radius of 30 ft that is intercepted by a central angle measuring 2 1/2 radians
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
2 1/2*30 ft=?
anonymous
  • anonymous
is that all you do?
anonymous
  • anonymous
no... what is the total circumference?

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anonymous
  • anonymous
360
anonymous
  • anonymous
C = 2*pi*r
anonymous
  • anonymous
so would it be... 360=2*pi*30? do I put the 360?
anonymous
  • anonymous
you don't... there are 360 degrees in a circle. there are 2*pi radians in a circle too. these are equivalent. the arc of the circle encompassed by 2 1/2 radians is proportional to the total \[\frac{ 2\frac{ 1 }{ 2 } }{2\pi } \text{ is the proportion of the circle's circumference covered by this \angle}\] so \[\text{Arc length }=\left(\frac{2\frac{1}{2}}{2\pi}\right)2\pi r =2\frac{1}{2}\pi r \]
anonymous
  • anonymous
thank you so much for your help! you made it a lot easier to understand :) could you please help me with 1 more problem?
anonymous
  • anonymous
yeah... what is it?
anonymous
  • anonymous
oops... i forgot to cancel the pi... it should be \[\text{Arc length }=2\frac{1}{2} r \]
anonymous
  • anonymous
okay thank you! and its, find the central angle of a sector in degrees if the area of the sector is 220 ft^2 and the radius of the circle is 12 feet.
anonymous
  • anonymous
So what's the area of a sector?
anonymous
  • anonymous
is it A=a/2*r^2?
anonymous
  • anonymous
yes, if a = angle in radians... so plug in what you know and solve for what you need...
anonymous
  • anonymous
if you want, you can do that witthout numbers and then plug them in... \[ A = \frac{\theta}{2}r^{2} \text{ solve for }\theta\]
anonymous
  • anonymous
so 220=a/2 (12)^2?
anonymous
  • anonymous
yeah, now solve for a...
anonymous
  • anonymous
do you divide by 24?
anonymous
  • anonymous
I mean 144
anonymous
  • anonymous
yes
anonymous
  • anonymous
so is the answer 3.05 repeated?
anonymous
  • anonymous
\[ 220=\frac{\theta}{2}(12)^{2}=\frac{\theta}{2}(144)=72\,\theta\Rightarrow \theta=\frac{220}{72}=\frac{55}{18}\]
anonymous
  • anonymous
thank you for having the patience for helping me! :)
anonymous
  • anonymous
thanks for persisting and learning!
anonymous
  • anonymous
:)

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