Question 155376
# 1


Let x = width of path
 

First let's find the area of the garden only: Area of Garden = 18*13=234
 

So the area of the garden only is 234 square feet.
 

Now let's draw a picture:


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/ziggy%20rayon/solution_visual_aid.jpg" alt="Photobucket - Video and Image Hosting">


From the picture, notice that the length of the entire rectangle (including the width of the path) is {{{18+2x}}} (notice there are two "x" lengths per side) and the total width is {{{13+2x}}} .
 
 
So the area of the entire enclosure (including the path) is the expression {{{(18+2x)(13+2x)}}}
 
 
{{{(18+2x)(13+2x)-234}}} Now subtract off the area of the garden (we only want the area of the path)
 

 
{{{234+36x+26x+4x^2-234}}} FOIL
 

 
{{{4x^2+62x}}} Combine like terms
 

 
So the area of the path only is {{{A=4x^2+62x}}}
 
 

{{{A=4x^2+62x}}} Start with the area of the path
 

 
{{{516=4x^2+62x}}} Plug in {{{A=516}}} (which is the area of the path)
 

 
{{{0=4x^2+62x-516}}} Subtract 516 from both sides



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=4}}}, {{{b=62}}}, and {{{c=-516}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(62) +- sqrt( (62)^2-4(4)(-516) ))/(2(4))}}} Plug in  {{{a=4}}}, {{{b=62}}}, and {{{c=-516}}}



{{{x = (-62 +- sqrt( 3844-4(4)(-516) ))/(2(4))}}} Square {{{62}}} to get {{{3844}}}. 



{{{x = (-62 +- sqrt( 3844--8256 ))/(2(4))}}} Multiply {{{4(4)(-516)}}} to get {{{-8256}}}



{{{x = (-62 +- sqrt( 3844+8256 ))/(2(4))}}} Rewrite {{{sqrt(3844--8256)}}} as {{{sqrt(3844+8256)}}}



{{{x = (-62 +- sqrt( 12100 ))/(2(4))}}} Add {{{3844}}} to {{{8256}}} to get {{{12100}}}



{{{x = (-62 +- sqrt( 12100 ))/(8)}}} Multiply {{{2}}} and {{{4}}} to get {{{8}}}. 



{{{x = (-62 +- 110)/(8)}}} Take the square root of {{{12100}}} to get {{{110}}}. 



{{{x = (-62 + 110)/(8)}}} or {{{x = (-62 - 110)/(8)}}} Break up the expression. 



{{{x = (48)/(8)}}} or {{{x =  (-172)/(8)}}} Combine like terms. 



{{{x = 6}}} or {{{x = -43/2}}} Simplify. 



So the answers are {{{x = 6}}} or {{{x = -43/2}}} 

  
Since a negative width doesn't make sense, this means that the only solution is {{{x=6}}}



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Answer:


So the width of the path can be up to 6 feet (ie 6 feet is the maximum width)