Question 1120368
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<pre>
Let the dimensions of the rectangular piece be x and y,

    where y be the height of the cylinder.

    (Hence, x is the dimension to be bent into circle/cylinder).


Then we have  <U>FIRST EQUATION</U>  for the original area of the piece 

    xy = 800     (1)     (square inches)


When the dimension "x" is bent to the circle (to the cylinder latent surface), its radius becomes  r = {{{x/(2*pi)}}}.


Then the volume of the cylinder is  

    V = {{{pi*r^2*y}}} = {{{pi*(x/(2*pi)^2)*y}}} = {{{pi*(x^2/(4*pi^2))*y}}} = {{{(x^2/(4*pi))*y}}}   cubic inches,


therefore our  <U>SECOND EQUATION</U> is

    {{{(x^2/(4*pi))*y}}} = 400,     (2)       

or

     {{{x^2*y}}} = {{{1600*pi}}}.     (2').


Thus we have the system of two equations


     xy = 800,          (1)

     {{{x^2*y}}} = {{{1600*pi}}}.    (2')


In  (2')  replace  xy  by  800,  based on (1).  You will get

     {{{800*x}}} = {{{1600*pi}}},    or

     x = {{{2*pi}}}.


Thus you solved for x.  Now substitute  x = {{{2*pi}}}  into (1) to get

     {{{2*pi*y}}} = 800.


Then you get  y = {{{800/(2*pi)}}} = {{{400/pi}}}.


<U>Answer</U>.   x = {{{2*pi}}} inches;   y = {{{400/pi}}} inches.
</pre>

Solved.


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<U>Be aware</U>:  &nbsp;&nbsp;the system written by &nbsp;@josgarithmetic&nbsp; in his post


{{{system(xy=800,y*(pi*x^2)=400)}}}


is &nbsp;&nbsp;<U>I N C O R R E C T</U> !



For your safety, simply ignore it !