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One Hundred Concentric Circles - Geometry

by Andrew
(New York)










































One hundred concentric circles with radii 1, 2, 3, …, 100 are drawn in a plane. The interior of the circle of radius one is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. What is the ratio of the total area of the green regions to the area of the largest circle?

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Mar 24, 2012
One Hundred Concentric Circles
by: Staff


Part I

Question:


by Andrew
(New York)

One hundred concentric circles with radii 1, 2, 3, …, 100 are drawn in a plane. The interior of the circle of radius one is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. What is the ratio of the total area of the green regions to the area of the largest circle?


Answer:

The ratio of the total area of the green regions to the area of the largest circle is shown below:

Ratio = (r + 1) / 2r
(the derivation of this formula is shown below)


Since there are 100 concentric circles

Ratio = (100 + 1) / (2*100)

Ratio = 101 / 200

>>> final answer: Ratio = 101 / 200

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Derivation of the formula: Ratio = (r + 1) / 2r

For this particular problem you can determine the ratio for any even value of the radius “r”.

RED - 1st circle - The interior of the center circle is RED.

GREEN - 2nd circle - The interior of the area between the center circle and the 2nd circle is GREEN.

RED - 3rd circle - The interior of the area between the 2nd circle and the 3rd circle is RED.

GREEN - 4th circle - The interior of the area between the 3rd circle and the 4th circle is GREEN.
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The interior of the area between every odd numbered circle (r = 1, 3, 5, 7, …) and the next concentric circle (which is even numbered) is GREEN.

The radius of every GREEN circle is an even number (r = 2, 4, 6, 8, …).



The area for a circle = πr²


1st circle, r = 1
area = π*1² = 1 π

2nd circle, r = 2
area = π*2² = 4 π

3rd circle, r = 3
area = π*3² = 9 π

4th circle, r = 4
area = π*4² = 16 π

5th circle, r = 5
area = π*5² = 25 π

6th circle, r = 6
area = π*6² = 36 π
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The GREEN area between any odd circle and the next highest (adjacent) even circle:

Green Area between Concentric Circles = {[area of next highest (adjacent) even circle] - (area of odd circle)}

Total Green Area = Sum of all Green Areas between Concentric Circles


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Mar 24, 2012
One Hundred Concentric Circles
by: Staff

------------------------------------------------

Part II


Circles 1 and 2


The GREEN area between circles 1 and 2:

Green_Area_1_2 = (area of circle 2) - (area of circle 1)

Green_Area_1_2 = 4 π - 1 π

Green_Area_1_2 = 3 π



Total Green Area = Sum of all Green Areas between Concentric Circles

Total Green Area = Green_Area_1_2

Total Green Area = 3 π


Ratio of green area between 1st and 2nd circles to the area of the area of the larges circle (2nd circle):

Ratio = [(Area of 2nd circle) - (Area of 1st circle)] / (Area of 2nd circle)

Ratio = (4π - 1π) / (4π)

Ratio = (3π) / (4π)

Ratio = 3 / 4

-----

Circles 3 and 4


The GREEN area between circles 3 and 4:

Green_Area_3_4 = (area of circle 4) - (area of circle 3)

Green_Area_3_4 = 16 π - 9 π

Green_Area_3_4 = 7 π



Total Green Area = Sum of all Green Areas between Concentric Circles

Total Green Area = Green_Area_1_2 + Green_Area_3_4

Total Green Area = 3 π + 7 π

Total Green Area = 10 π





Ratio of total green area to the area of the area of the largest circle (4th circle):

Ratio = (10 π) / (16 π)

Ratio = 10 / 16

Ratio = 5 / 8

-----

Circles 5 and 6


The GREEN area between circles 5 and 6:

Green_Area_5_6 = (area of circle 6) - (area of circle 5)

Green_Area_5_6 = 36 π - 25 π

Green_Area_5_6 = 11 π



Total Green Area = Sum of all Green Areas between Concentric Circles

Total Green Area = Green_Area_1_2 + Green_Area_3_4 + Green_Area_5_6

Total Green Area = 3 π + 7 π + 11 π

Total Green Area = 21 π



Ratio of total green area to the area of the area of the largest circle (6th circle):

Ratio = (21 π) / (36 π)

Ratio = 21 / 36

Ratio = 7 / 12

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A general formula can be derived from this pattern:

For r = even number (only)

Ratio = (r + 1) / (2r)

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Thanks for writing.

Staff
www.solving-math-problems.com



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