Question 195654
{{{P=2L+2W}}} Start with the perimeter formula



{{{14=2L+2W}}} Plug in {{{P=14}}}



{{{7=L+W}}} Divide every term by 2



{{{7-W=L}}} Subtract W from both sides.



{{{L=7-W}}} Rearrange the terms.



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{{{a^2+b^2=c^2}}} Move onto the Pythagorean Theorem



{{{L^2+W^2=5^2}}} Plug in {{{a=L}}}, {{{b=W}}}, and {{{c=5}}}



{{{(7-W)^2+W^2=5^2}}} Plug in {{{L=7-W}}}



{{{(7-W)^2+W^2=25}}} Square 5 to get 25



{{{49-14W+W^2+W^2=25}}} FOIL



{{{49-14W+W^2+W^2-25=0}}} Subtract 25 from both sides.



{{{2W^2-14W+24=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{aW^2+bW+c}}} where {{{a=2}}}, {{{b=-14}}}, and {{{c=24}}}



Let's use the quadratic formula to solve for W



{{{W = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{W = (-(-14) +- sqrt( (-14)^2-4(2)(24) ))/(2(2))}}} Plug in  {{{a=2}}}, {{{b=-14}}}, and {{{c=24}}}



{{{W = (14 +- sqrt( (-14)^2-4(2)(24) ))/(2(2))}}} Negate {{{-14}}} to get {{{14}}}. 



{{{W = (14 +- sqrt( 196-4(2)(24) ))/(2(2))}}} Square {{{-14}}} to get {{{196}}}. 



{{{W = (14 +- sqrt( 196-192 ))/(2(2))}}} Multiply {{{4(2)(24)}}} to get {{{192}}}



{{{W = (14 +- sqrt( 4 ))/(2(2))}}} Subtract {{{192}}} from {{{196}}} to get {{{4}}}



{{{W = (14 +- sqrt( 4 ))/(4)}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}. 



{{{W = (14 +- 2)/(4)}}} Take the square root of {{{4}}} to get {{{2}}}. 



{{{W = (14 + 2)/(4)}}} or {{{W = (14 - 2)/(4)}}} Break up the expression. 



{{{W = (16)/(4)}}} or {{{W =  (12)/(4)}}} Combine like terms. 



{{{W = 4}}} or {{{W = 3}}} Simplify. 



So the widths are {{{W = 4}}} or {{{W = 3}}}



If we plug these widths into {{{L=7-W}}}, we'll get the lengths


{{{L=3}}} or {{{L=4}}}



So we'll make the length the longer dimension. 



So the length is 4 m and the width is 3 m