SOLUTION: A goat is tied outside the fence of a 5-m diameter circular garden using a 3-m rope. One end of the rope is tied to the collar around the neck of the goat and the other s attached

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Question 1050269: A goat is tied outside the fence of a 5-m diameter circular garden using a 3-m rope. One end of the rope is tied to the collar around the neck of the goat and the other s attached to the fence at the point which is at the level as the goat's collar (the goat can move anywhere within 3 meters from the outside of the circular fence). Find the area by which the goat can move around by integration.
Found 2 solutions by Fombitz, ikleyn:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
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The red circle is the circular garden.
The purple circle is the 3m radius from the edge of the circular garden fence.
The equations of the two circles are,
x%5E2%2By%5E2=%282.5%29%5E2
1.x%5E2%2By%5E2=6.25
%28x-2.5%29%5E2%2By%5E2=%283%29%5E2
%28x-2.5%29%5E2%2By%5E2=9
To find the intersection points substitute from eq. 1 into eq. 2,
y%5E2=6.25-x%5E2
So,
%28x-2.5%29%5E2%2B6.25-x%5E2=9
x%5E2-5x%2B6.25%2B6.25-x%5E2=9
x%5E2-5x%2B12.5-x%5E2=9
-5x=-3.5
x=0.7
So then integrate to get the sliver area (shown in green),
the limits of integration would be x=0.7 and x=2.5.
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A=int%28%28y%5B2%5D-y%5B1%5D%29%2Cdx%2Cx=0.7%2C2.5%29
A=int%28%28sqrt%289-%28x-2.5%29%5E2%29-sqrt%286.25-x%5E2%29%29%2Cdx%2Cx=0.7%2C2.5%29
A=1.874 <-- Obtained numerically
Double that plus half of the area of the large circle will be the complete area that the goat can graze,
A%5Btot%5D=2%281.874%29%2B%28pi%283%29%5E2%29%2F2
A%5Btot%5D=17.89

Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
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A goat is tied outside the fence of a 5-m diameter circular garden using a 3-m rope. One end of the rope is tied to the collar
around the neck of the goat and the other s attached to the fence at the point which is at the level as the goat's collar
(the goat can move anywhere within 3 meters from the outside of the circular fence).
Find the area by which the goat can move around by integration.
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Once you got the intersection points for two circles (in the solution of the other tutor),
the remaining part is to subtract the areas of two segments of the circle from the area of the larger circle.

You do not need to calculate integrals.

You need to know the formula for the area of a segment of a circle.

See the lessons
    - Area of a segment of the circle
    - Solved problems on area of a circle,
    - Solved problems on area of a segment of the circle
    - Solved problems on area of a circle, a sector and a segment of the circle
in this site related to this topic.

You have this free of charge online textbook on Geometry
GEOMETRY - YOUR ONLINE TEXTBOOK
in this side.

The referred lessons are the part of this online textbook under the topic "Area of a circle, of a sector and of a segment of a circle".