Question 183395
Let L=length and W=width


Since the "land is 70 feet longer than it is wide", this tells us that {{{L=W+70}}} (ie take the width and add 70 to get the length)



If you cut the rectangle in half along the diagonal, you'll find that the resulting pieces will be right triangles. So we can use Pythagorean's Theorem to find the legs (since we're given the hypotenuse)



{{{a^2+b^2=c^2}}} Start with Pythagorean's Theorem



{{{L^2+W^2=130^2}}} Plug in {{{a=L}}}, {{{b=W}}} and {{{c=130}}} (this is the given diagonal)



{{{L^2+W^2=16900}}} Square 130 to get 16900



{{{(W+70)^2+W^2=16900}}} Plug in {{{L=W+70}}}



{{{W^2+140W+4900+W^2=16900}}} FOIL



{{{W^2+140W+4900+W^2-16900=0}}} Subtract 16900 from both sides.



{{{2W^2+140W-12000=0}}} Combine like terms.



Notice how the equation is now in the form {{{ax^2+bx+c}}} where {{{a=2}}}, {{{b=140}}}, and {{{c=-12000}}}



{{{W = (-140 +- sqrt( (140)^2-4*2*-12000 ))/(2*2)}}} Plug in {{{a=2}}}, {{{b=140}}}, and {{{c=-12000}}}




{{{W = (-140 +- sqrt( 19600-4*2*-12000 ))/(2*2)}}} Square 140 to get 19600




{{{W = (-140 +- sqrt( 19600+96000 ))/(2*2)}}} Multiply {{{-4*-12000*2}}} to get 96000




{{{W = (-140 +- sqrt( 115600 ))/(2*2)}}} Combine like terms in the radicand (everything under the square root)




{{{W = (-140 +- 340)/(2*2)}}} Simplify the square root 



{{{W = (-140 +- 340)/4}}} Multiply 2 and 2 to get 4




So now the expression breaks down into two parts



{{{W = (-140 + 340)/4}}} or {{{x = (-140 - 340)/4}}}



Lets look at the first part:


{{{W=(-140 + 340)/4}}}



{{{W=200/4}}} Add the terms in the numerator



{{{W=50}}} Divide



So one possible answer is {{{W=50}}}




Now lets look at the second part:


{{{W=(-140 - 340)/4}}}



{{{W=-480/4}}} Subtract the terms in the numerator



{{{W=-120}}} Divide



So another possible answer is {{{W=-120}}}



However, it doesn't make sense to have a negative width. So we must ignore this possible solution




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So this means that the only answer is {{{W=50}}} and that the width is 50 feet.



{{{L=W+70}}} Go back to the first equation



{{{L=50+70}}} Plug in {{{W=50}}}



{{{L=120}}} Add



So the other answer is {{{L=120}}} which means that the length is 120 feet.



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Answer:



So the length is 120 feet and the width is 50 feet.