maths 難題高手請解 20分

Circumference of a circle = pi x 2 x radius

22/7(2r) - 3.14159(2r) = 1
6.28571429r – 6.28318531r = 1

r = 100
c = 22/7 (2x100) = 629 (round it off to whole number)
c = 3.14159 (2x100) = 628 (round it off to whole number)
r = 99
c = 22/7 (2x99) = 622 (round it off to whole number)
c = 3.14159 (2x99) = 622 (round it off to whole number)

For circumference

Question (1)
The smallest whole number value of r is 100 cm for which the two answers are different.

Question (2)
The largest whole number value of r is 99 cm for which the two answers are the same.


Volume of sphere = 4/2 x pi x radius^3
(4/3)(22/7)(r)^3 - (4/3)(3.14159)(r)^3 = 1
4.19047619(r)^3 – 4.18879021(r)^3 = 1

r = 10
V = (4/3)(22/7)(10)^3 = 4.19047619 x 1000 = 4190 cm^3 (round it off to whole number)
V = (4/3)( 3.14159)(10)^3 = 4.18879021x 1000 = 4189 cm^3 (round it off to whole number)
r = 9
V = (4/3)(22/7)(9)^3 = 4.19047619 x 729 = 3055 cm^3 (round it off to whole number)
V = (4/3)( 3.14159)(9)^3 = 4.18879021x 729 = 3054 cm^3 (round it off to whole number)
r = 8
V = (4/3)(22/7)(8)^3 = 4.19047619 x 512 = 2145 cm^3 (round it off to whole number)
V = (4/3)( 3.14159)(8)^3 = 4.18879021x 512 = 2145 cm^3 (round it off to whole number)

Question (3)
For volume of sphere

The smallest whole number value of r is 9 cm for which the two answers are different.

The largest whole number value of r is 8 cm for which the two answers are the same.

1. The smallest whole number value of r for which the two answers are different is 30. Proven using the identity 2(7x+2)/7=(2x+4/7) =(x+0.571428...)
2(22/7)(r)-2(pi)(r)>4/7-1/2
=1/14
r>1/[28(22/7-pi)]>=29
The smallest (7x+2) which is equal to or greater than 29 is 30, where x is a natural number.
2(22/7)(30)=1320/7=188.571428......~189
2(pi)(30)=60pi=188.4955592......~188

2. The largest whole number value of r for which the two answers are equal is . Proven using the identity 2(7x+5)/7=(2x+1+3/7)
When 2(22/7)(r) - 2(pi)(r)=(1+3/7)-(1/2), r=338.93
r<=338
The greatest (7x+5) whic his equal to or smaller than 338 is 334, where x is a natural number.
2(22/7)(334)=14696/7=2099.428571......~2099
2(pi)(334)=668pi=2098.583893......~2099

3. The smallest whole number value of r for which the two answers are equal is 6, and can be easily found out by trial and error.
The largest whole number value of r is 8, as illustrated by Godfrey.

2011-01-18 14:50:30 補充：
對不起…昨天小弟匆忙中計算出錯
2. When 2(22/7)(r)-2(pi)(r)=(1+3/7)-(1/2), r=338.93 <---應為367.17
r<=367
The greatest (7x+5) which is equal to or smaller than 367 is 362, where x is a natural number.
2(22/7)(362)=15928/7=2275.428571......~2275
2(pi)(362)=724pi=2274.513081......~2275

2011-01-22 12:11:49 補充：
更詳細解釋，請參閱小弟"Pi (Exact) and 22/7"系列文章（共三篇）
1st Paper: http://hk.knowledge.yahoo.com/question/article?qid=6911012001968
2nd Paper: http://hk.knowledge.yahoo.com/question/article?qid=6911012001975
3rd Paper: http://hk.knowledge.yahoo.com/question/article?qid=6911012001978