Question 1152412
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<pre>

Let x and y be the values under the question, i.e. dimensions of the original piece of cardboard.


The height of the box is 2 cm;

the area of the base of the box is equal to its volume divided by the height

    area of the base of the box = {{{504/2}}} = 252 cm^2.


It gives you your first equation


    (x-2*2)*(y-2*2) = 252,   or

    (x-4)*(y-4) = 252.       (1)


The second equation is for the surface area of the box

    (x-4)*(y-4) + 2*(x-4) + 2*(y-4) = 396 - 4*(2*2)  cm^2,   or

    (x-4)*(y-4) + 2*(x-4) + 2*(y-4) = 380.    (2)


It is your second equation.


Simplify them.


In the left side of (2),  replace  (x-4)*(y-4) by 252, based on (1).

You will get then instead of (2)

    252 + 2*(x-4) + 2*(y-4) = 380,   or

    2*(x-4) + 2*(y-4) = 380 - 252 = 128,

    (x-4) + (y-4) = 64.


So, the sum  (x-4) + (y-4) = 64,  while the product  (x-4)*(y-4) = 252.


Thus (x-4) and (y-4) are the roots of the quadratic equation  t^2 - 64t + 252 = 0.


Hence,  {{{t[1,2]}}} = {{{(64 +- sqrt(64^2 - 4^252))/2}}} = {{{(64 +- sqrt(3088))/2}}} = {{{(64 +- 4*sqrt(193))/2}}} = {{{32 +- 2*sqrt(193))}}}.


It gives  x-4 = {{{32 + 2*sqrt(193)}}},  x = {{{36+2*sqrt(193)}}} = 63.785 cm.

          y-4 = {{{32 - 2*sqrt(193)}}},  y = {{{36-2*sqrt(193)}}} =  8.215 cm.


<U>CHECK</U>.  Equation (1) : (x-4)*(y-4) = (63.785 -4)*(8.215-4) = 251.994 = 252 cm^2;

        Equation (2) : (x-4)*(y-4) + 2*(x-4) + 2*(y-4) = (63.785 -4)*(8.215-4) + 2*(63.785 -4) + 2*(8.215-4) = 379.994 = 380 cm^2.


<U>ANSWER</U>.  The dimensions are  x = {{{36+2*sqrt(193)}}} = 63.785 cm  and  y = {{{36-2*sqrt(193)}}} =  8.215 cm.
</pre>

Solved.


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If you want to see many other similar solved problems, look into the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Surface-area/Making-a-box-from-a-piece-of-cardboard.lesson>Making a box from a piece of cardboard</A> 

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this online textbook under the topic 
"<U>Dimensions and the area of rectangles and circles and their elements</U>".


Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.