Question

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

Answer 1

2 1/2*30 ft=?

Answer 2

is that all you do?

Answer 3

no... what is the total circumference?

Answer 4

360

Answer 5

C = 2*pi*r

Answer 6

so would it be... 360=2*pi*30? do I put the 360?

Answer 7

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

\[\text{Arc length }=\left(\frac{2\frac{1}{2}}{2\pi}\right)2\pi r =2\frac{1}{2}\pi r \]

Answer 8

thank you so much for your help! you made it a lot easier to understand :) could you please help me with 1 more problem?

Answer 9

yeah... what is it?

Answer 10

oops... i forgot to cancel the pi... it should be

\[\text{Arc length }=2\frac{1}{2} r \]

Answer 11

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.

Answer 12

So what's the area of a sector?

Answer 13

is it A=a/2*r^2?

Answer 14

yes, if a = angle in radians...

so plug in what you know and solve for what you need...

Answer 15

if you want, you can do that witthout numbers and then plug them in...

\[ A = \frac{\theta}{2}r^{2} \text{ solve for }\theta\]

Answer 16

so 220=a/2 (12)^2?

Answer 17

yeah, now solve for a...

Answer 18

do you divide by 24?

Answer 19

I mean 144

Answer 20

yes

Answer 21

so is the answer 3.05 repeated?

Answer 22

\[ 220=\frac{\theta}{2}(12)^{2}=\frac{\theta}{2}(144)=72\,\theta\Rightarrow \theta=\frac{220}{72}=\frac{55}{18}\]

Answer 23

thank you for having the patience for helping me! :)

Answer 24

thanks for persisting and learning!

Answer 25

:)