Question 93440
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(3,0,20),
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 20 to get 20+2x. This is the total length of the side that contains 20 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. 


So that means the total area can be written as:


{{{A=(20+2x)(30+2x)}}}




Now lets find the inner area:

{{{A=20*30=600}}} plug in the garden dimensions

So the inner area is 600 sq ft



Now lets find the total area

inner area+remaining area=600+400=1000



So the total area is 1000 sq ft





{{{1000=(20+2x)(30+2x)}}} Plug in {{{A=1000}}} (this is the total area)



{{{1000=600+100x+4x^2}}} Foil


{{{0=600+100x+4x^2-1000}}} Subtract 1000 from both sides


{{{0=4x^2+100x-400}}} Combine like terms




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+100*x-400=0}}} ( notice {{{a=4}}}, {{{b=100}}}, and {{{c=-400}}})


{{{x = (-100 +- sqrt( (100)^2-4*4*-400 ))/(2*4)}}} Plug in a=4, b=100, and c=-400




{{{x = (-100 +- sqrt( 10000-4*4*-400 ))/(2*4)}}} Square 100 to get 10000  




{{{x = (-100 +- sqrt( 10000+6400 ))/(2*4)}}} Multiply {{{-4*-400*4}}} to get {{{6400}}}




{{{x = (-100 +- sqrt( 16400 ))/(2*4)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (-100 +- 20*sqrt(41))/(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 = (-100 +- 20*sqrt(41))/8}}} Multiply 2 and 4 to get 8


So now the expression breaks down into two parts


{{{x = (-100 + 20*sqrt(41))/8}}} or {{{x = (-100 - 20*sqrt(41))/8}}}



Now break up the fraction



{{{x=-100/8+20*sqrt(41)/8}}} or {{{x=-100/8-20*sqrt(41)/8}}}



Simplify



{{{x=-25 / 2+5*sqrt(41)/2}}} or {{{x=-25 / 2-5*sqrt(41)/2}}}



So these expressions approximate to


{{{x=3.50781059358212}}} or {{{x=-28.5078105935821}}}



So our possible solutions are:

{{{x=3.50781059358212}}} or {{{x=-28.5078105935821}}}



Since a negative length doesn't make sense, our only solution is {{{x=3.50781059358212}}} which is 3.507 to the nearest thousandth





-----------------------------

Check:


{{{1000=(20+2x)(30+2x)}}} Start with the given area function


{{{1000=(20+2(3.508))(30+2(3.507))}}} plug in x=3.507


{{{1000=(20+7.016)(30+7.014)}}} multiply


{{{1000=(27.016)(37.014)}}} Add


{{{1000=999.970224}}} Since we rounded, this is as close as it gets. So our answer is verified.