Question 90931
If we draw a picture we get


{{{drawing(500,500,-5,5,-5,5,
line(-5,5,5,5),
line(5,5,5,-5),
line(5,-5,-5,-5),
line(-5,-5,-5,5),

arrow(-3,3,-5,3),
arrow(-5,3,-3,3),
locate(-4,3,x),

arrow(3,3,5,3),
arrow(5,3,3,3),
locate(4,3,x),


arrow(3,3,3,5),
arrow(3,5,3,3),
locate(3,4,x),


arrow(3,-3,3,-5),
arrow(3,-5,3,-3),
locate(3,-4,x),


locate(0,3,30),
locate(3,0,40),
rectangle(-3,-3,3,3)


)}}}


If we let x be the width of the path, notice there are 2 x-values per side. So we add 2x to 40 to get 40+2x. This is the total length of the side that contains 40 feet. 


Also, this means we add 2x to 30 to get 30+2x. This is the total length of the side that contains 30 feet. 



Start with the general area function:

{{{A=xy}}}


{{{1800=xy}}} Plug in {{{A=1800}}} (this is the total area)



{{{1800=(40+2x)(30+2x)}}} Plug in {{{x=40+2x}}} and {{{y=30+2x}}}



{{{1800=4x^2+140x+1200}}} Foil



{{{0=4x^2+140x-600}}} Subtract 1800 from both sides




Let's use the quadratic formula to solve for x:



Starting with the general quadratic


{{{ax^2+bx+c=0}}}


the general solution using the quadratic equation is:


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


So lets solve {{{4*x^2+140*x-600=0}}} ( notice {{{a=4}}}, {{{b=140}}}, and {{{c=-600}}})


{{{x = (-140 +- sqrt( (140)^2-4*4*-600 ))/(2*4)}}} Plug in a=4, b=140, and c=-600




{{{x = (-140 +- sqrt( 19600-4*4*-600 ))/(2*4)}}} Square 140 to get 19600  




{{{x = (-140 +- sqrt( 19600+9600 ))/(2*4)}}} Multiply {{{-4*-600*4}}} to get {{{9600}}}




{{{x = (-140 +- sqrt( 29200 ))/(2*4)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (-140 +- 20*sqrt(73))/(2*4)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{x = (-140 +- 20*sqrt(73))/8}}} Multiply 2 and 4 to get 8


So now the expression breaks down into two parts


{{{x = (-140 + 20*sqrt(73))/8}}} or {{{x = (-140 - 20*sqrt(73))/8}}}



Now break up the fraction



{{{x=-140/8+20*sqrt(73)/8}}} or {{{x=-140/8-20*sqrt(73)/8}}}



Simplify



{{{x=-35 / 2+5*sqrt(73)/2}}} or {{{x=-35 / 2-5*sqrt(73)/2}}}



So these expressions approximate to


{{{x=3.86000936329383}}} or {{{x=-38.8600093632938}}}



So our possible solutions are:

{{{x=3.86000936329383}}} or {{{x=-38.8600093632938}}}


Since a negative length doesn't make sense, our only solution is {{{x=3.86000936329383}}}


which is 3.860 to the nearest thousandth



Check:


{{{1800=(40+2x)(30+2x)}}} Start with the given area function


{{{1800=(40+2(3.860))(30+2(3.860))}}} plug in x=3.860


{{{1800=(40+7.72)(30+7.72)}}} multiply


{{{1800=(47.72)(37.72)}}} Add


{{{1800=1799.9984}}} Since we rounded, this is as close as it gets. So our answer is verified.