Question 893193
How wide is really the diameter, which is two times the radius, r.
Total surface area for the cylinder, assuming top and bottom are included and ignoring any openings is {{{(2pi*r)*20+2(pi*r^2)=679}}}.


Simplifying that,
{{{40pi*r+2pi*r^2=679}}}
{{{2pi*r^2+40pi*r-679=0}}}


General solution of quadratic equation, for r:
{{{r=(-40pi+sqrt(1600pi^2+4*2pi*679))/(4pi)}}}
{{{r=(-40pi+2sqrt(400pi^2+2pi*679))/(4pi)}}}
{{{highlight(r=-10+(1/(2pi))sqrt(400pi^2+2pi*679))}}}, or as whatever equivalent form you need.


Further, you wanted width, which is twice r; so you have width
{{{highlight(-20+(1/pi)sqrt(400pi^2+2pi*679))}}}