```Question 110067

A piece of wire having a total length of 72 cm was cut into two unequal segments and bent to form two unequal squares. If the total area of the squares is 180-sq.cm.,what is the difference in the lengths of the two segments?

{{{Area_of_Square1= x^2}}}

{{{Area_of_Square2= y^2}}}

{{{a _total _length_ of _wire = 72 cm}}}=> {{{ P1 + P2 = 72cm}}}

{{{the_ total_ area _of_ the squares = 180cm^2}}}

if {{{ P1 + P2 =72cm}}}..=> {{{4x + 4y = 4(x + y) = 72cm}}}then

=>{{{x+y = (72/4)cm}}}

=>{{{x+y = 18cm}}}

=>{{{x= 18cm - y}}}.. substitute in formula of  total area

{{{the_ total_ area _of_ the squares = 180cm^2}}}
{{{ the_ total_ area _of_ the squares = Area_of_Square1 + Area_of_Square2 }}}

{{{180cm^2= x^2 + y^2}}}

{{{180cm^2= (18cm - y )^2 + y^2}}}

{{{180cm^2= 18^2cm^2 - 36 ycm + y ^2 + y^2}}}

{{{180cm^2 - 324cm^2 =  - 36 ycm + y ^2 + y^2}}}

{{{-144cm^2 =  - 36 ycm + y ^2 + y^2}}}.move {{{-144cm^2}}} to the right

{{{0 =  2y ^2 - 36 ycm + 144cm^2 }}}.divide both sides by {{{2}}}

{{{0 =  y ^2 - 18 ycm + 72cm^2 }}}.

Or

{{{ y ^2 - 18 ycm + 72cm^2 = 0 }}}.

{{{y[1,2]=(-b +- sqrt (b^2 -4*a*c )) / (2*a)}}}

{{{y[1,2]=(-(-18) +- sqrt ((-18)^2 -4*1*72 )) / (2*1)}}}

{{{y[1,2]=(18 +- sqrt (324 - 288)) / 2}}}

{{{y[1,2]=(18 +- sqrt (36)) / 2}}}

{{{y[1,2]=(18 +- 6) / 2}}}

We need only positive root

{{{y[1]=(18 + 6) / 2}}}

{{{y[1]= 24 / 2}}}

{{{y[1]= 12cm}}}

Then {{{x}}} is:

{{{x= 18cm - y}}}..

{{{x= 18cm  12cm}}}..

{{{x= 6cm}}}..

Perimeter of each square {{{P1}}} and {{{P2}}} is:

{{{P1 = 4 x cm}}}

{{{P1 = 4 *6 cm}}}

{{{P1 = 24 cm}}}

And

{{{P2 = 4ycm}}}

{{{P2 = 4*12cm}}}

{{{P2 = 48cm}}}
Check:

{{{a _total _length_ of _wire = 72 cm}}}

{{{P1 + P2 = 72cm}}}

=> {{{24cm + 48cm = 72cm}}}

=> {{{72cm = 72cm}}}

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