All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Step Function II (Posted on 2011-08-06) Difficulty: 2 of 5
A step function f(x) is defined on the interval [0, 1)

From [0, 2/3), the step is 4/5 high.

From [2/3, 8/9), the step is 24/25 high.

From [8/9, 26/27), the step is 124/125 high.

From [1 - 3-k, 1 - 3-(k+1)), the step is 1 - 5-(k+1) high.

What is the exact area under the step function?

See The Solution Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer solution with a high degree of confidence | Comment 1 of 2
    5   point 7
   10   for K=0 to 75
   20    Wdth=3.0^(-K)-3.0^(-(K+1))
   30    Hgt=1-5.0^(-(K+1))
   40    Area=Wdth*Hgt
   50    Tot=Tot+Area
   60    print Area,Tot
   70   next
finds
  
 0.533333333333333333333333333333333     0.533333333333333333333333333333333
 0.213333333333333333333333333333333     0.746666666666666666666666666666666
 0.073481481481481481481481481481481     0.820148148148148148148148148148147
 0.024651851851851851851851851851851     0.8448
 0.008227818930041152263374485596707     0.853027818930041152263374485596707
 0.002743308641975308641975308641975     0.855771127572016460905349794238682
 0.000914483036122542295381801554641     0.856685610608139003200731595793323
 0.000304830800182898948331047096479     0.856990441408321902149062642889802
 0.000101610474825991972768378804044     0.857092051883147894121831021693847
 0.000033870172148554590255550475028     0.857125922055296448712086572168876
 0.000011290058307733125596256216588     0.857137212113604181837682828385464
 0.00000376335283090314823282358719      0.857140975466435084985915651972654
 0.000001254450947744967613212630063     0.857142229917382829953528864602717
 0.000000418150316189028195666750087     0.857142648067699018981724531352805
 0.000000139383438747945331306419366     0.857142787451137766927055837772171
 0.000000046461146250533061507751078     0.85714283391228401746011734552325
 0.000000015487048750258883907402222     0.857142849399332767719001252925473
 0.000000005162349583425041085010865     0.857142854561682351144042337936338
 0.000000001720783194475374569395407     0.857142856282465545619416907331746
 0.000000000573594398158482248091254     0.857142856856059943777899155423
 0.00000000019119813271949568658327      0.857142857047258076497394842006271
 0.000000000063732710906498669120169     0.85714285711099078740389351112644
 0.000000000021244236968832896835106     0.857142857132235024372726407961546
 0.00000000000708141232294429942026      0.857142857139316436695670707381807
 0.000000000002360470774314766505101     0.857142857141676907469985473886909
 0.000000000000786823591438255503812     0.857142857142463731061423729390721
 0.000000000000262274530479418501411     0.857142857142726005591903147892133
 0.000000000000087424843493139500479     0.857142857142813430435396287392613
 0.000000000000029141614497713166827     0.85714285714284257204989400055944
 0.000000000000009713871499237722275     0.857142857142852285921393238281716
 0.000000000000003237957166412574091     0.857142857142855523878559650855808
 0.00000000000000107931905547085803      0.857142857142856603197615121713838
 0.00000000000000035977301849028601      0.857142857142856962970633611999848
 0.000000000000000119924339496762003     0.857142857142857082894973108761851
 0.000000000000000039974779832254        0.857142857142857122869752941015852
 0.000000000000000013324926610751333     0.857142857142857136194679551767186
 0.000000000000000004441642203583777     0.857142857142857140636321755350963
 0.000000000000000001480547401194592     0.857142857142857142116869156545556
 0.000000000000000000493515800398197     0.857142857142857142610384956943753
 0.000000000000000000164505266799398     0.857142857142857142774890223743152
 0.000000000000000000054835088933132     0.857142857142857142829725312676285
 0.00000000000000000001827836297771      0.857142857142857142848003675653996
 0.000000000000000000006092787659236     0.857142857142857142854096463313233
 0.000000000000000000002030929219745     0.857142857142857142856127392532978
 0.000000000000000000000676976406581     0.85714285714285714285680436893956
 0.000000000000000000000225658802193     0.857142857142857142857030027741754
 0.000000000000000000000075219600731     0.857142857142857142857105247342485
 0.000000000000000000000025073200243     0.857142857142857142857130320542728
 0.000000000000000000000008357733414     0.857142857142857142857138678276143
 0.000000000000000000000002785911138     0.857142857142857142857141464187281
 0.000000000000000000000000928637046     0.857142857142857142857142392824327
 0.000000000000000000000000309545681     0.857142857142857142857142702370009
 0.000000000000000000000000103181894     0.857142857142857142857142805551903
 0.000000000000000000000000034393964     0.857142857142857142857142839945868
 0.000000000000000000000000011464654     0.857142857142857142857142851410523
 0.000000000000000000000000003821551     0.857142857142857142857142855232074
 0.00000000000000000000000000127385      0.857142857142857142857142856505925
 0.000000000000000000000000000424616     0.857142857142857142857142856930542
 0.000000000000000000000000000141539     0.857142857142857142857142857072081
 0.000000000000000000000000000047179     0.85714285714285714285714285711926
 0.000000000000000000000000000015726     0.857142857142857142857142857134987
 0.000000000000000000000000000005242     0.857142857142857142857142857140229
 0.000000000000000000000000000001747     0.857142857142857142857142857141977
 0.000000000000000000000000000000582     0.857142857142857142857142857142559
 0.000000000000000000000000000000194     0.857142857142857142857142857142753
 0.000000000000000000000000000000064     0.857142857142857142857142857142818
 0.000000000000000000000000000000021     0.85714285714285714285714285714284
 0.000000000000000000000000000000007     0.857142857142857142857142857142847
 0.000000000000000000000000000000002     0.857142857142857142857142857142849
 0.0     0.85714285714285714285714285714285
 0.0     0.85714285714285714285714285714285
 0       0.85714285714285714285714285714285
 0       0.85714285714285714285714285714285
 0       0.85714285714285714285714285714285
 0       0.85714285714285714285714285714285
 0       0.85714285714285714285714285714285
 


then, manually, as the decimal obviously repeats,

? 8//10 + 571428//9999990
 6//7
 
so the answer is 6/7.


  Posted by Charlie on 2011-08-07 12:50:30
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (7)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information