數學問題 PLZZ

A = area, R = radius of sphere, pi = 3.14159, * means multiply
n = 1 + % increase (in decimal)
Surface Area of sphere = 4*pi*(Radius)^2
A = 4*pi*(R)^2
New area of sphere = 1.12 of old area of sphere (Area increases by 12%)
1.12*4*pi*(R)^2 = 4*pi*(nR)^2
1.12*(R)^2 = (nR)^2
1.12*(R)^2 = (n)^2(R)^2
1.12 = (n)^2
n = square root of 1.12
n = 1.0583
% increase of radius = (n – 1)x100% = 5.83%

Vold = volume of old sphere (舊體積)
Vnew = volume of new sphere (新體積)
Volume of sphere = (4/3)*pi*(Radius)^3

Vold = (4/3)*pi*R^3
Vnew = (4/3)*pi*(1.0583R)^3
% increase in volume = (Vnew – Vold)/Vold * 100%

% increase in volume =
[(4/3)*pi*(1.0583R)^3 -(4/3)*pi*R^3]
---------------------------------------------- * 100%
(4/3)*pi*R^3

(4/3)*pi*[(1.0583r)^3 -r^3]
---------------------------------- * 100%
(4/3)*pi*r^3

(4/3)*pi*[(1.0583)^3r^3 -r^3]
---------------------------------- * 100%
(4/3)*pi*r^3

(4/3)*pi*r^3[(1.0583)^3 -1]
---------------------------------- * 100%
(4/3)*pi*r^3

1.0583)^3 -1
--------------------- * 100%
1

1.1852948 -1
--------------------- x 100%
1

0.1852948
--------------- * 100% = 18.53%
1

半徑增加 5.83%
體積增加18.53%

Check:
如果半徑是 (radius) = 1 m
舊半徑 = 1 m,
舊面積 = 4*pi*(1)^2= 12.56637 m^2, 舊體積 =(4/3)*pi*(1)^3 = 4.1888 m^3

新半徑= 1.0583 m,
新面積 = 4*pi*(1.0583)^2= 14.074321 m^2, 新體積 =(4/3)*pi*(1.0583)^3 = 4.9649 m^3

%體積增加 = (14.074321 – 12.56637)/12.56637 * 100% = 12%
%體積增加 =(4.9649 – 4.18888)/4.1888 * 100% = 18.53%

半徑增加 5.83%
體積增加18.53%

Check:
如果半徑是 (radius) = 1 m
舊半徑 = 1 m,
舊面積 = 4*pi*(1)^2= 12.56637 m^2, 舊體積 =(4/3)*pi*(1)^3 = 4.1888 m^3

新半徑= 1.0583 m,
新面積 = 4*pi*(1.0583)^2= 14.074321 m^2, 新體積 =(4/3)*pi*(1.0583)^3 = 4.9649 m^3

%體積增加 = (14.074321 – 12.56637)/12.56637 * 100% = 12%
%體積增加 =(4.9649 – 4.18888)/4.1888 * 100% = 18.53%

i recommend using easy and short calculation to do a simple question.

http://upload.trend.hk/images/1307700192.bmp
圖片參考：http://upload.trend.hk/images/1307700192.bmp

半徑 = (開方0.12) *100%
體積 = [(開方0.12)^3 ] *100%

^3 = 3次方