Question 151681
Let x=width and y=length


Remember, the perimeter of any rectangle is:


{{{P=2x+2y}}}



{{{14=2x+2y}}} Plug in {{{P=14}}} (the given perimeter)



{{{14-2x=2y}}} Subtract {{{2x}}} from both sides.



{{{2y=-2x+14}}} Rearrange the terms



{{{y=-x+7}}} Divide both sides by 2 to isolate y



Through the use of pythagoreans theorem, we get:



{{{x^2+y^2=5^2}}}



{{{x^2+y^2=25}}} Square 5 to get 25



{{{x^2+(-x+7)^2=25}}} Plug in {{{y=-x+7}}}



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



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



{{{2x^2-14x+24=0}}} Combine like terms. Note: {{{x^2+x^2<>x^4}}}



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



Let's use the quadratic formula to solve for x



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



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



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



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



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



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



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



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



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



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



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



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

  
{{{y=-x+7}}} Go back to the second equation



{{{y=-4+7}}} Plug in {{{x = 4}}}



{{{y=3}}} Add



So if the width is 4 inches, the length is 3 inches



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{{{y=-x+7}}} Go back to the second equation



{{{y=-3+7}}} Plug in {{{x = 3}}}



{{{y=4}}} Add



So if the width is 3 inches, the length is 4 inches



Note: the length is usually the longer of the two. So your book probably has the width of 3 inches and a length of 4 inches