SOLUTION: The area of the shaded region represents one-third of the total, What is the size of the radii such that this is possible? The radius of the whole circle is 12cm while the sha

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Question 1182220: The area of the shaded region represents one-third of the total, What is the
size of the radii such that this is possible?
The radius of the whole circle is 12cm while the shaded region is a ring in
the middle with a small not shaded circle in the middle.
Found 4 solutions by ankor@dixie-net.com, Edwin McCravy, greenestamps, MathTherapy:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
he area of the shaded region represents one-third of the total, What is the size of the radii such that this is possible?
The radius of the whole circle is 12cm while the shaded region is a ring in the middle with a small not shaded circle in the middle.
;
Find 1/3 of the area of the whole circle
1%2F3*pi%2A12%5E2+=+150.796
let R = radius to the outer ring
let r = radius to the inner ring
:
%28pi%2AR%5E2%29+-+%28pi%2Ar%5E2%29+=+150.796
pi%28R%5E2+-+r%5E2%29+=+150.796
divide both sides by pi
R%5E2+-+r%5E2+=+48
graph this equation
R+=+sqrt%2848%2Br%5E2%29 Where R is on the y axis, r is on the x axis
first integer solution
R = 8, r = 4
see if that works
%28pi%2A8%5E2%29+-+%28pi%2A4%5E2%29
201.062 - 50.265 = 150.797
The radii of 8 and 4 make this possible

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!


Since the green area is 1/3 of the area of the outer circle, the area of the
inner circle is 2/3 of the area of the outer circle.

Using the area of a circle formula A=pi%2Ar%5E2,

pi%2Ar%5E2=expr%282%2F3%29pi%2A12%5E2

cross%28pi%29%2Ar%5E2=expr%282%2F3%29cross%28pi%29%2A12%5E2

r%5E2=expr%282%2F3%2912%5E2

r%5E2=expr%282%2F3%29%28144%29

r%5E2=96

r=sqrt%2896%29

r=sqrt%2816%2A6%29

r=4sqrt%286%29

So the radius of the inner circle must be 4√6 or about 9.8 cm.

Edwin

Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


The area of the ring is 1/3 the area of the circle with radius 12cm, so the area of the inner circle is 2/3 the area of the large circle.

Since the ratio of the areas of the two circles is 2:3, the ratio of the radii of the two circles is the square root of that ratio -- sqrt(2):sqrt(3).

ANSWER: The radius of the inner circle is

12%2A%28sqrt%282%29%2Fsqrt%283%29%29 = 9.7979..., or 9.78 to 2 decimal places.

Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!

The area of the shaded region represents one-third of the total, What is the
size of the radii such that this is possible?
The radius of the whole circle is 12cm while the shaded region is a ring in
the middle with a small not shaded circle in the middle. 
Area of larger circle: matrix%281%2C5%2C+pi%2Ar%5E2%2C+%22=%22%2C+pi%2812%29%5E2%2C+%22=%22%2C+144pi%29

With radius of smaller circle being r, we get the area of the smaller circle as: pi%2Ar%5E2

Area of shaded region: 144%2Api+-+pi%2Ar%5E2

Area of shaded region is also 1%2F3 the area of the larger circle.

We then get: matrix%281%2C3%2C+%281%2F3%29+%2A+144pi%2C+%22=%22%2C+144%2Api+-+pi%2Ar%5E2%29

           

                     

Since this is a measurement, CORRECT answer is highlight_green%284sqrt%286%29%29, as the negative value for the radius is IGNORED.