Jump to content
BrainDen.com - Brain Teasers
  • 0

Sum of circumferences


Go to solution Solved by bonanova,

Question

5 answers to this question

Recommended Posts

  • 0
  • Solution

Diameter D1 of the incircle is some fraction f of the triangle's height h.

The next circle has a diameter D2 that is the same fraction f of the remaining height h2 = h - D1.

And so forth. Here, since h = 12 and D1 = 20/3, we have f = 5/9.

The successive diameters are:

f h                          =                       h     =  6.666666667

f (1 - h f)                =                  f - h f2     =  2.962962963

f (1 - f + h f2 )         =             ff2 + h f3   =  1.316872428

f (1 - f + f2 - h f3)   =    f  -  f2+ f3 - h f4    =  0.585276635

  

and so on.

 

The corresponding areas are 34.90658504, 6.895127909, 1.362000575, 0.269037151 ...

Taking the first 50 terms, the diameters sum to 12.00000000

And the areas sum to 43.4989752.

 

If we note that successive diameters decrease by a factor of (1 - f)

Then the areas of successive circles decrease by a factor of r = (1 - f)2.

Then the areas sum to A1/(1 - r).

 

That's just pi (10/3)2/[1 - (4/9)2] = 34.9066/.80247 = 43.4989

 

Link to post
Share on other sites
  • 0

For area, had to check the web on how to calculate the radius of incircle
So, the first circle has radius 10/3
So the first circle covers 2/3 of the altitude length
Without going into details, I would say by symmetry that each circle covers the next 2/3rd
So the second circle would be of radius = 1/3(10 - 20/3) = 10/9
ratio of each consecutive circle radius = 1/3
So sum of areas = pi(r1² + r2² + ...)
= pi.(10/3)² / (1 - 1/9) = (25/4)pi

Link to post
Share on other sites
  • 0

For area, had to check the web on how to calculate the radius of incircle

So, the first circle has radius 10/3

So the first circle covers 2/3 of the altitude length

Without going into details, I would say by symmetry that each circle covers the next 2/3rd

So the second circle would be of radius = 1/3(10 - 20/3) = 10/9

ratio of each consecutive circle radius = 1/3

So sum of areas = pi(r1² + r2² + ...)

= pi.(10/3)² / (1 - 1/9) = (25/4)pi

The first circle has radius 10/3 but doesn't cover 2/3 of the altitude length

Link to post
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Answer this question...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...
  • Recently Browsing   0 members

    No registered users viewing this page.

×
×
  • Create New...