Question 1203311
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        The solution by @MathLover1 is incorrect.
        I came to bring you a correct solution.



Surface area of a sphere: 

{{{A=4*pi*r^2}}}


let altitude be {{{h}}} and diameter {{{d}}}
given ratio is {{{1:5}}}


so, {{{h/d=1/5}}}


{{{h=d/5}}}

{{{r=d/2 }}} &nbsp;&nbsp;--> &nbsp;&nbsp;{{{h=(2r)/5}}}  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<<<---===  &nbsp;&nbsp;it is where &nbsp;@MathLover1 &nbsp;made her mistake.


The area {{{A}}} of a spherical zone can be calculated using the formula 


{{{A=2*pi*rh }}}

where {{{h }}}is the height of the spherical zone and {{{r}}} is the radius of the sphere.


if  the area of the  spherical zone  is  {{{80pi}}},  we have

{{{80pi=2*pi*rh}}}...simplify

{{{40=rh}}}.....substitute {{{h}}}

{{{40=r*(2r)/5)}}}

{{{100=r^2}}}

{{{r=sqrt(100)}}}

{{{r=10}}}


then, surface area of a sphere is: 

{{{A=4*pi*10^2}}}

{{{A=4*pi*100}}}

{{{A=400*pi}}}    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>ANSWER</U>


Solved &nbsp;&nbsp;(correctly).