Question 118426
If you draw a picture, you can see that the dimensions of the bottom of the box are {{{20-2x}}} by {{{10-2x}}}



<a href="http://s150.photobucket.com/albums/s91/jim_thompson5910/?action=view&current=box2.jpg" target="_blank"><img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/box2.jpg" border="0" alt="Photobucket"></a>



So the area of the bottom is {{{A=(20-2x)(10-2x)}}}



{{{96=(20-2x)(10-2x)}}} Now plug in the given area {{{A=96}}}



{{{96=200-60x+4x^2}}} Foil



{{{0=200-60x+4x^2-96}}} Subtract 96 from both sides



{{{0=104-60x+4x^2}}} Combine like terms



{{{0=4x^2-60x+104}}} Rearrange the terms



Let's use the quadratic formula to solve for x:



Starting with the general quadratic


{{{ax^2+bx+c=0}}}


the general solution using the quadratic equation is:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}




So lets solve {{{4*x^2-60*x+104=0}}} ( notice {{{a=4}}}, {{{b=-60}}}, and {{{c=104}}})





{{{x = (--60 +- sqrt( (-60)^2-4*4*104 ))/(2*4)}}} Plug in a=4, b=-60, and c=104




{{{x = (60 +- sqrt( (-60)^2-4*4*104 ))/(2*4)}}} Negate -60 to get 60




{{{x = (60 +- sqrt( 3600-4*4*104 ))/(2*4)}}} Square -60 to get 3600  (note: remember when you square -60, you must square the negative as well. This is because {{{(-60)^2=-60*-60=3600}}}.)




{{{x = (60 +- sqrt( 3600+-1664 ))/(2*4)}}} Multiply {{{-4*104*4}}} to get {{{-1664}}}




{{{x = (60 +- sqrt( 1936 ))/(2*4)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (60 +- 44)/(2*4)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{x = (60 +- 44)/8}}} Multiply 2 and 4 to get 8


So now the expression breaks down into two parts


{{{x = (60 + 44)/8}}} or {{{x = (60 - 44)/8}}}


Lets look at the first part:


{{{x=(60 + 44)/8}}}


{{{x=104/8}}} Add the terms in the numerator

{{{x=13}}} Divide


So one answer is

{{{x=13}}}




Now lets look at the second part:


{{{x=(60 - 44)/8}}}


{{{x=16/8}}} Subtract the terms in the numerator

{{{x=2}}} Divide


So another answer is

{{{x=2}}}


So our possible solutions are:

{{{x=13}}} or {{{x=2}}}


However, since 13 is greater than 10, the solution {{{x=13}}} is not possible since the square cutout cannot have a longer length than one of the sides.



So our only solution is {{{x=2}}}




------------

Check:



Remember, the area is


{{{A=(20-2x)(10-2x)}}}



{{{96=(20-2(2))(10-2(2))}}} Plug in the given area {{{A=96}}} and {{{x=2}}}



{{{96=(20-4)(10-4)}}} Multiply



{{{96=(16)(6)}}} Subtract



{{{96=96}}} Multiply. Since the two sides of the equation are equal, this verifies our answer.