The Chamfered Box

Geometry Level 5

Let there be a solid characterized by the equation ( x a ) 2.5 + ( y b ) 2.5 + ( z c ) 2.5 < 1. { \left( \frac { x }{ a } \right) }^{ 2.5 }+{ \left( \frac { y }{ b } \right) }^{ 2.5 } + { \left( \frac { z }{ c } \right) }^{ 2.5 }<1.

Calculate the volume of this solid if a = b = 2 a = b =2 and c = 3 c = 3 .


The answer is 60.8591.

Steven Zheng by Steven Zheng , 26, China

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1 solution

Ronak Agarwal
Dec 8, 2014

My method is the computer programming method that is point picking method.

I generate a random point ( x , y , z ) (x,y,z) where 0 ≤ x , y ≤ 2 0 \le x,y \le 2 and 0 ≤ z ≤ 3 0 \le z \le 3 and check for each try whether the equation at the left side is less than 1 or not.

After 40000000 40000000 trials we have got total of 25356280 25356280 points inside our solid

Hence volume of our solid in first quadrant = P o i n t s i n s i d e t h e s o l i d P o i n t s i n s i d e t h e c u b o i d × V o l u m e o f C u b o i d \frac{Points\quad inside\quad the\quad solid}{Points\quad inside\quad the\quad cuboid} \times Volume\quad of\quad Cuboid

Calculating we get = 12 × 0.633907 = 7.6068 =12 \times 0.633907 = 7.6068

Hence the entire volume is = 8 ∗ 7.6068 = 60.854 = 8*7.6068 = 60.854

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Interesting. I am not familiar with the point picking method. How do you program that? I would like to learn. If you want to know how I solved this equation, you can visit this note I wrote Dirichlet Triple Integral .

Steven Zheng - 6 years, 6 months ago

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Here's my code in Java 

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public static void main(String[] args) {

      int k=0 ;
      for(int j=0 ; j<40000000;j=j+1){
      if(isgood())
          k=k+1 ;
      }
      System.out.print(k) ;
    }

I created a method isgood() whose code is as follows : 

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public static boolean isgood(){
  boolean k ;
  Random rn = new Random() ;
  double x = 2*rn.nextDouble() ;
  double y = 2*rn.nextDouble() ;
  double z = 3*rn.nextDouble() ;
  double ans = Math.pow((x/2),2.5)+Math.pow((y/2),2.5)+
  Math.pow((z/3),2.5) ;
  k = ans < 1 ;
  return k ;

  }

And the result was as follows 

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25356280 BUILD SUCCESSFUL (total time: 28 seconds)

Ronak Agarwal - 6 years, 6 months ago

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can you please tell me what is the function of the statement you wrote Random rn=new Random() ; and the nextDouble(); method.

Gaurav Jain - 6 years, 4 months ago

Essentially, he ran a monte carlo simulation by picking serval points ( x , y , z ) (x, y, z) and seeing if they lay inside the solid.

Calvin Lin Staff - 6 years, 5 months ago

Can you please explain how did you get 0 ≤ x , y ≤ 2 0 \le x,y \le 2 and 0 ≤ z ≤ 3 0 \le z \le 3 i.e. how you decided on the upper limits of 2 and 3 ?

Kudou Shinichi - 6 years, 4 months ago

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