Question 1202100
<font color=black size=3>
Answer: <font color=red>14 ft by 36 ft</font>



Work Shown:


x = width
y = height


P = perimeter of the rectangle
P = 2*(width + height)
P = 2*(x + y)
Replace P with 100 as this is the perimeter, aka amount of fencing
100 = 2(x + y)
100/2 = x + y
50 = x + y
y = 50-x


A = area of the rectangle
A = width*height
A = x*y
A = x*(50-x)
A = 50x-x^2
A = -x^2+50x
Plug in A = 504 which is the desired area.
504 = -x^2+50x
0 = -x^2+50x-504
-x^2+50x-504 = 0
-(x^2-50x+504) = 0
x^2-50x+504 = 0


You could take a trial-and-error approach to factoring, but I think the quadratic formula is more efficient.
Plug in a = 1, b = -50, c = 504
{{{x = (-b +- sqrt(b^2 - 4ac))/(2a)}}}


{{{x = (-(-50) +- sqrt((-50)^2 - 4(1)(504)))/(2(1))}}}


{{{x = (50 +- sqrt(2500-2016))/(2(1))}}}


{{{x = (50 +- sqrt(484))/(2)}}}


{{{x = (50 +- 22)/(2)}}}


{{{x = (50 + 22)/(2) }}} or {{{ x = (50 - 22)/(2)}}}


{{{x = (72)/(2) }}} or {{{ x = (28)/(2)}}}


{{{x = 36 }}} or {{{ x = 14}}}


If x = 36, then y = 50-x = 50-36 = 14
If x = 14, then y = 50-x = 50-14 = 36


The rectangle is <font color=red>14 ft by 36 ft</font>


Check:
Perimeter = 2*(14+36) = 2*50 = 100
Area = 14*36 = 504
The answers are confirmed.
</font>