Question 1008424
{{{drawing(300,225,-1,50,-1,37.25,
rectangle(0,0,50,36),
rectangle(4,4,46,32),
locate(23,32,42ft),
locate(4.5,20,28ft),
green(arrow(6,32,6,36)),
green(arrow(6,36,6,32)),
locate(6.5,35.5,green(w))
)}}}
With length in ft and areas in square feet,
{{{system(pool_area=42*28,total_area=(42+2w)*(28+2w),border_area=624)}}}-->{{{(42+2w)*(28+2w)-42*28=624}}}-->{{{42*28+42*(2w)+(2w)*28+(2w)*(2w)-42*28=624}}}-->{{{42*28+84w+56w+4w^2-42*28=624}}}-->{{{140w+4w^2=624}}}-->{{{35w+w^2=156}}}
We can solve the quadratic equation {{{35w+w^2=156}}} by any method
("completing the square", factoring, or using the quadratic formula),
to get the solutions
{{{w=4}}} and {{{w=-36}}} .
For this word problem, the solution {{{w=-36}}} does not make sense,
so the maximum with for the border (achieved using all the tile available) is {{{highlight(4ft)}}} .
 
SOLVING THE EQUATION BY COMPLETING THE SQUARE:
{{{w^2+35w=156}}}
{{{w^2+35w+(35/2)^2-(35/2)^2=156}}}
{{{(w+35/2)^2-1225/4=156}}}
{{{(w+35/2)^2=1225/4+156}}}
{{{(w+35/2)^2=1225/4+624/4}}}
{{{(w+35/2)^2=1849/4}}}
{{{(w+35/2)^2=(43/2)^2}}}-->{{{system(w+35/2=43/2,"or",w+35/2=-43/2)}}}-->{{{system(w=43/2-35/2,"or",w=-43/2-35/2)}}}-->{{{system(w=8/2,"or",w=-78/2)}}}-->{{{system(w=4,"or",w=-36)}}}