Question

Two sources of sound are located on the x axis, and each emits power uniformly in all directions. There are no reflections. One source is positioned at the origin and the other at x = +116 m. The source at the origin emits four times as much power as the other source. Where on the x axis are the two sounds equal in intensity? Note that there are two answers.

xbetween the sources =
xnot between the sources =

Please provide answer and explain solution.

Answer 1

    Given

two sources of sound 1 is at origin and the another one is at 116 m

so the distance between two sources is 116 m

The source at the origin emits four times as much power as the other source

              P1 = 4 P2    

at x distance from the source at origin the intensity of two sources are equal

               I1 = I2

the relation between power and intensity is

             I = P / A    

            P1 / A1   = P2 / A2  

            4 P2   / A1 = P2 / A2

              4 / 4πr₁²   = 1 / 4π r₂²   

                  4 / r₁² = 1 / r₂²     

                 4 / x ² = 1 / ( 116 - x ) ²  

                   2 / x   = 1 / 116 - x

                    2 ( 116 - x ) = x

                      232 = 3 x

                      x = 77.33 m ( between the sources )

at x = 77.33 m the intensity of two sounds are equal    

   Out side the sources

            

I = P / A    

            P1 / A1   = P2 / A2  

            4 P2   / A1 = P2 / A2

              4 / 4πr₁²   = 1 / 4π r₂²   

                  4 / r₁² = 1 / r₂²     

                 4 / x ² = 1 / ( 116 + x ) ²  

                   2 / x   = 1 / 116 + x

                    2 ( 116 + x ) = x

                     x = -232 m

    negative sign represents opposite direction to x- axis                  

Answer 2

    Given

two sources of sound 1 is at origin and the another one is at 126 m

so the distance between two sources is 126 m

The source at the origin emits four times as much power as the other source

              P1 = 4 P2    

at x distance from the source at origin the intensity of two sources are equal

               I1 = I2

the relation between power and intensity is

             I = P / A    

            P1 / A1   = P2 / A2  

            4 P2   / A1 = P2 / A2

              4 / 4πr₁²   = 1 / 4π r₂²   

                  4 / r₁² = 1 / r₂²     

                 4 / x ² = 1 / ( 126 - x ) ²  

                   2 / x   = 1 / 126 - x

                    2 ( 126 - x ) = x

                      252 = 3 x

                      x = 84 m ( between the sources )

at x = 84 m the intensity of two sounds are equal    

   Out side the sources

I = P / A    

            P1 / A1   = P2 / A2  

            4 P2   / A1 = P2 / A2

              4 / 4πr₁²   = 1 / 4π r₂²   

                  4 / r₁² = 1 / r₂²     

                 4 / x ² = 1 / ( 126 + x ) ²  

                   2 / x   = 1 / 126 + x

                    2 ( 126 + x ) = x

                     x = -252 m

    negative sign represents opposite direction to x- axis