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#1 2006-06-29 02:21:48

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Sector Perimeter

I have a question here asking for the Perimeter of a shaded section of a circle:

The full circle is shown with center O.  One sector is labled with a radius of 20 and angle Pi/2 (90 degrees).  The rest of the circle is shaded and that is the part I need to work out the perimeter of.
I know that a full circle is 2Pi radians (360 degrees) and since the unshaded sector is Pi/2 radians I can deduce
2Pi - Pi/2 = 3Pi/2 (270 degrees)

Now I just use the regular formula for working out the arc length which is
s = rθ
(let θ = the angle in radians)
s = 20 * 3Pi/2 = 60Pi/2 = 94.2 (3.s.f)

Is this correct? The answers at the back of the book states134.2!


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#2 2006-06-29 02:24:24

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Sector Perimeter

Do I need to set my calculator to radian mode?  How do I do this? It's a casio fx-83wa.


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#3 2006-06-29 02:37:01

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Sector Perimeter

Notice something.  Your answer is off by 134.2 - 94.2 = 40.  Coincidence?  I think not.  That number should sound familiar.  What piece are you missing?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#4 2006-06-29 02:40:55

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Sector Perimeter

40? Hum... I can see that the first sector (with the angle Pi/2) has an arc length of 10Pi and the other sector has an arc of 30Pi, that adds up to 40Pi  but I do not see how it is related?

P.S. This is the second question I've ever done of this module so I do not know much about it wink

Last edited by rickyoswaldiow (2006-06-29 02:42:34)


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#5 2006-06-29 02:45:19

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Sector Perimeter

I could look at it this way:
The first sector has angle Pi/2 and radius 20. s=rθ=20*Pi/2=10Pi

Since Pi/2 = 90 degrees which is 1/4 of the full circle, the shaded sector is the other 3/4 of the circle.  1/4 of the circle = 10Pi and thus 3/4 of it would = 30Pi, 94.2.


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#6 2006-06-29 02:47:51

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Sector Perimeter

40 = 2*20 = 2*r.  You have to not only measure the arc length but also the...


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#7 2006-06-29 02:50:16

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Sector Perimeter

Aha! Of course smile
The perimeter stretches along the two straight edges of the unshaded sector and not just round the arc of the shaded area.


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#8 2006-06-29 05:32:08

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Sector Perimeter

Right!


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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