Question 257059
cutting equal squares from each corner of an 11 in. by 14 in. piece of cardboard.
 the area of the bottom must be 80 in^2.
 what size should she cut from each corner
:
x = side of the cutout squares
:
The bottom dimensions of the box will be (11-2x) by (14-2x)
;
The area of the bottom given as 80 sq/in, therefore:
;
(11-2x)*(14-2x) = 80
FOIL
154 - 22x - 28x + 4x^2 = 80
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 solve this:
{{{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 = 10.78, obviously, this solution is not valid
and
{{{x = (25 - 18.138)/(4) }}}
x = {{{6.862/4}}}
x = 1.7 inches, the side of the cutout square
:
:
Check solution on a calc
Enter (11 - 2(1.7)) * (14 - 2(1.7)) = 80.56 ~ 80 sq/in