Question 186088
 make an open-top box by cutting equal squares from the four corners of
 an 11 inch by 14 inch sheet of a cardboard and folding up the sides.
 If the area of the base is to be 80 square inches then what size square
 should be cut from each corner?
;
Let x = the side of the square to cut from each corner
:
Draw a rough diagram of what is described, labeling the sides of the squares
as x and rectangular sheet as 11 by 14. It will be apparent to you that the
dimensions of the bottom of the box will be (11-2x) by (14-2x), which is given
as an area of 80 sq/in
:
(11-2x) * (14-2x) = 80
FOIL
154 - 22x - 28x + 4x^2 = 80
Arrange as a quadratic equation:
4x^2 - 50x + 154 - 80 = 0
:
4x^2 - 50x + 74 = 0
Simplify, divide by 2:
2x^2 - 25x + 37 = 0
:
Use the quadratic formula to find x
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
in this problem a=2, b=-25, c=37
{{{x = (-(-25) +- sqrt(-25^2 - 4 * 2 *37 ))/(2*2) }}}
:
{{{x = (25 +- sqrt(625 - 296 ))/(4) }}} 
:
{{{x = (25 +- sqrt(329))/(4) }}}
Two solutions
{{{x = (25 + 18.138)/(4) }}} 
x = {{{43.138/4}}}
x = 10.784
and
{{{x = (25 - 18.138)/(4) }}} 
x = {{{6.862/4}}}
x = 1.712 inch squares,  This is the only solution that makes sense here.
:
:
Check solution. 2 * 1.712 = 3.43, subtract from the original dimensions and
find the area
(11-3.43) * (14-3.43) =
7.57 * 10.57 = 80.0 sq/in confirms our solution