Question 198597
{{{50*40 = 2000}}} cm2
Let the edge of the corner squares = {{{x}}}
After the squares in each corner are cut, the area left is
{{{2000 - 4x}}}
The areas of the sides of the box will be
{{{x*(50 - 2x)}}} (times 2) and
{{{x*(40 - 2x)}}} (times 2)
What is left is the base, so
{{{2000 - 4x^2 = 2x*(50 - 2x) + 2x*(40 - 2x) + 875}}} cm2
{{{2000 - 4x^2 = 100x - 4x^2 + 80x - 4x^2 + 875}}}
{{{4x^2 - 180x + 1125 = 0}}}
Use quadratic equation
 {{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
{{{a = 4}}}
{{{b = -180}}}
{{{c = 1125}}}
{{{x = (-(-180) +- sqrt( (-180)^2-4*4*1125 ))/(2*4) }}}
{{{x = ( 180 +- sqrt( 32400 - 18000 ))/8 }}}  
{{{x = ( 180 +- sqrt( 14400 ))/8 }}} 
{{{x = ( 180 +- 120)/8 }}}
{{{x = 60/8}}}
{{{x = 7.5}}} cm
and
{{{x = 300/8}}}
{{{x = 37.5}}} cm (this is impossible- it's too long)
The side length of a square is 7.5 cm
check answer:
{{{2000 - 4x^2 = 2x*(50 - 2x) + 2x*(40 - 2x) + 875}}} cm2
{{{2000 - 4*7.5^2 = 2*7.5*(50 - 2*7.5) + 2*7.5*(40 - 2*7.5) + 875}}}
{{{2000 - 4*56.25 = 15*(50 - 15) + 15*(40 - 15) + 875}}}  
{{{2000 - 225 = 15*35 + 15*25 + 875}}}  
{{{1775 = 525 + 375 + 875}}}
{{{1775 = 1775}}}
OK
The volume of the box would be:
{{{x*(50 - 2x)*(40 - 2x)}}}
{{{7.5*(50 - 15)*(40 - 15)}}}
{{{7.5*35*25}}}
{{{6562.5}}} cm3
The volume is 6562.5 cm3
check:
Volume is also
{{{875*x}}}
{{{875*7.5 = 6562.5}}}
OK