Question 150955
{{{A = 192}}} in2
{{{P = 56}}} in
Let {{{l}}} = length
Let {{{w}}}= width
{{{l*w = 192}}}
{{{w = 192/l}}}
{{{2l + 2w = 56}}}
{{{2l + 2*(192/l) = 56}}}
{{{2l + 384/l = 56}}}
multiply both sides by {{{l}}}
{{{2l^2 + 384 = 56l}}}
{{{2l^2 - 56l + 384 = 0}}}
divide both sides by {{{2}}}
{{{l^2 - 28l + 192 = 0}}}
Solve by completing the square
{{{l^2 - 28l = -192}}}
{{{l^2 - 28l + (28/2)^2 = -192 + (28/2)^2}}}
{{{l - 14)^2 = 196 - 192}}}
{{{l - 14)^2 = 4}}}
Take the square root of both sides
{{{l - 14 = 2}}}
{{{l = 16}}}
{{{l - 14 = -2}}}
{{{l = 12}}}
I'll graph it to check
{{{ graph( 700, 600, -5, 20, -5, 10, x^2 - 28x + 192) }}}
Looks like the length can be either {{{16}}} or {{{12}}}
If {{{l=16}}},
{{{l*w = 192}}}
{{{w = 192/16}}}
{{{w = 12}}}
The length is 16 and the width is 12 (answer)
If {{{l = 12}}}
{{{w = 192/12}}}
{{{w = 16}}} This is really the same answer with {{{w >l}}}
which it shouldn't be
check answer:
{{{2l + 2w = 56}}}
{{{2*16 + 2*12 = 56}}}
{{{32 + 24 = 56}}}
{{{56 = 56}}}
OK