Question 98653
Question:

Find the dimensions of a rectangle whose area is 180cm^2 and whose perimeter is 54 cm.

Answer:

Given that, 

{{{Area = 180 cm^2 }}} and 

{{{ perimeter = 54 cm }}}


Perimeter of rectangle is given by the formula, 2( length + width)


==> 2( length + width) = 54


==>  ( length + width) = {{{ 54/2}}}


==> ( length + width) = 27


Now suppose, length = x cm


Then width = 27 - x 



Now area is given by the formula, length * width



That is length * width = 180 cm^2


==> x * ( 27-x) = 180


==> {{{ 27x - x^2 = 180 }}}


==> {{{ x^2 - 27 x + 180 = 0 }}} -----------------(1) 


This is a quadratic equation and you can solve this equation using quadratic formula for x.



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



Comparing equation(1) with the standard quadratic equation, 
{{{ax^2 + bx + c = 0}}}, we have,


a = 1, b = -27 and c = 180


==> {{{x = (-(-27) +- sqrt((-27)^2-4*1*180 ))/(2*1) }}}


==> {{{x = ( 27 +- sqrt(729-720)) / 2  }}}


==> {{{x = ( 27 +- sqrt 9 ) /2  }}}


==> {{{x = ( 27 + - 3)/2 }}}



==> either, {{{ x = (27 + 3)/2}}} or {{{ x = (27 - 3)/2}}}



==> either, {{{ x = 30 /2}}} or {{{ x = 24/2}}}



==> either, {{{ x = 15}}} or {{{ x = 12}}}



that is either length = 15 cm of 12 cm


When length = 15 cm, width =  27 - 15 = 12 cm



And when length = 12cm, width = 27 - 12 = 15 cm.


Hence the solution.



Hope you found the explanation useful.


Regards.


Praseena.