A cylindrical hole of radius xis bored through a shere of radius R in such a way that the axis of the hole passes throiugh the centerof the sphere. Find the value of x that maximizes the complete surface area of the remaining solid. Hint: The area of a segment of height h on a shere of radius R is 2 π R h

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A cylindrical hole of radius xis bored through a shere of radius R in such a way that the axis of the hole passes throiugh the centerof the sphere. Find the value of x that maximizes the complete surface area of the remaining solid. Hint: The area of a segment of height h on a shere of radius R is   2  π  R  h

Leonard Mahoney · Accepted Answer

Step 1Please use the given hint. The surface area of a segment of height h on a sphere of radius R is   2  π  R  h . Here   h  =  R  −            R      2        −          x      2       . There are two such segments removed by drilling the hole.So surface area removed from the sphere:  4  π      R    2    −  4  π  R            R      2        −          x      2      Surface area of the sphere remaining:  4  π  R            R      2        −          x      2      Surface area of the cylindrical surface through the sphere:  2  π  x  ⋅  2            R      2        −          x      2      Total surface   S  =  4  π  R            R      2        −          x      2        +  4  π  x            R      2        −          x      2        S  =  4  π  (  R  +  x  )            R      2        −          x      2                          d        S                    d        x              =                  4        π        (                  R          2                −        R        x        −        2                  x          2                )                              R          2                −                  x          2                    Solving       R    2    −  R  x  −  2      x    2    =  0 , we find that S is maximized when   x  =  R      /    2

A cylindrical hole of radius xis bored through a shere of radius R in such a way that the axis of th

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