concentric circles with radii, outside and inside areas

[aaaaaa1234]

i need help doing this, i can't figure it out. and there are too many circles.

 

[royhaas]

So you are looking at the difference between areas of circles with radii \(\displaystyle 2k\) and \(\displaystyle 2k-1\), for \(\displaystyle k = 1,2, ... 24\).

 

[Mrspi]

i need help doing this, i can't figure it out. and there are too many circles.

Here's another approach...start with some MUCH SIMPLER problems, and see if you can see a pattern that will lead you to a result for the complex problem.

What if there are just TWO circles....radius 1 for the smaller one, and radius 2 for the larger. What's the area between the two circles?

area of big circle - area of small circle = pi*2[sup:2ll1lhjl]2[/sup:2ll1lhjl] - pi*1[sup:2ll1lhjl]2[/sup:2ll1lhjl] = 4 pi - pi, or 3 pi

What if there are FOUR circles, with radii 1, 2, 3, and 4. You already know that the area between the first two circles is 3 pi. Find the area between the third and fourth circles, and add that to what you already have: 3pi + [pi*4[sup:2ll1lhjl]2[/sup:2ll1lhjl] - pi*3[sup:2ll1lhjl]2[/sup:2ll1lhjl]}, or 3pi + (16 pi - 9 pi), or 3 pi + 7 pi, or 10 pi

With 2 circles, area is 3 pi
With 4 circles, area is 10 pi

Continue this process...

 

[Mrspi]

i need help doing this, i can't figure it out. and there are too many circles.

Here's another approach...start with some MUCH SIMPLER problems, and see if you can see a pattern that will lead you to a result for the complex problem.

What if there are just TWO circles....radius 1 for the smaller one, and radius 2 for the larger. What's the area between the two circles?

area of big circle - area of small circle = pi*2[sup:21ce2br5]2[/sup:21ce2br5] - pi*1[sup:21ce2br5]2[/sup:21ce2br5] = 4 pi - pi, or 3 pi

What if there are FOUR circles, with radii 1, 2, 3, and 4. You already know that the area between the first two circles is 3 pi. Find the area between the third and fourth circles, and add that to what you already have: 3pi + [pi*4[sup:21ce2br5]2[/sup:21ce2br5] - pi*3[sup:21ce2br5]2[/sup:21ce2br5], or 3pi + (16 pi - 9 pi), or 3 pi + 7 pi, or 10 pi

With 2 circles, area is 3 pi
With 4 circles, area is 10 pi

Continue this process...

 

[TchrWill]

i need help doing this, i can't figure it out. and there are too many circles.

If I understand you correctly, the circle of radius 1 is blank.
The area between the circle of radius 2 and radius 1 is shaded.
The area between the circle of radius 3 and radius 2 is blank.
The area between the circle of radius 4 and radius 5 is shaded.

This continues out to the final radius of 48.
What is the sum total of the shaded areas?

n............1.........2.........3.........4..........5
R..........1-2......3-4......5-6......7-8......9-10
Area.....3Pi....7Pi...11Pi...15Pi...19Pi
Sum......3Pi...10Pi...21Pi...36Pi...55Pi

Diff..........7Pi.....11Pi.....15Pi.....19Pi
Diff...............4Pi........4Pi........4Pi

We have a finite difference series where the 2nd differences are constant at 4 Pi making the general expression for any sum of the form an^2 + bn + c = S

Using the derived data:
a(1)2 + b(1) + c = 3 or a + b + c = 3
a(2)^2 + b(2) + c = 10 or 4a + 2b + c = 10
a(3)^2 + b(3) + c = 21 or 9a^2 + 3b + c = 21

Solving, a = 2, b = 1 and c = 0 yielding S = 2n^2 + n.

Since n = R/2, we can rewrite it as S = sum = R(R + 1)/2.

Therefore, the total of the shaded areas covering the the gaps from an odd radius circle and an even radius circle becomes

S = 48(48 + 1)/2 = 1,176 square units.