Question 626145
A farmer wants to build a pen in the following shape to keep his ostriches and emu separate.  The pen for the ostriches is a square and the pen for the emu is a rectangle which has the length 10 feet more than its width.  <font color = "red">The ostriches need 4.5 times as much area as the emus.</font>  If the total amount of fencing is 160 feet, find the outside dimensions of the pen.
<pre>
Note: I had to make up the shape of the pens, and I had to make up the 
sentence in red because it can't be worked with only the information you gave.

{{{drawing(400,400,-3, 43,-3,43,

line(0,0,30,0), line(30,0,30,30), line (0,30,30,30),line(0,0,0,40),
line(0,40,20,40), locate(15,0,x), locate(31,15,x), locate(10,42,y+10),
locate(21,36,y),locate(-1.5,15,x),locate(-1.5,36,y), locate(11,16,OSTRICHES), 
line(20,40,20,30),locate(8,36,EMUS) )}}}
 
Area of ostrich pen = side² = x²
Area of emu pen = = length·width = y·(y+10)

Area of ostrich pen = 4.5(area of emu pen)

                 x² = 4.5y·(y+10)

Fencing = 4 sides of square + top of rectangle + 2 vertical sides of rectangle
    
    160 = 4x + (y+10) + 2y

So we have this system:

 x² = 4.5y·(y+10)
160 = 4x + (y+10) + 2y

Simplifying the first:

 x² = 4.5y²+45y

Simplifying the second:

 160 = 4x + y + 10 + 2y
 160 = 4x + 3y + 10
 150 = 4x + 3y

Solve for x
 150 - 3y = 4x
 {{{(150-3y)/4}}} = x

Substitute in  x² = 4.5y²+45y

 x² = 4.5y²+45y
 x² = 4.5y²+45y
 {{{((150-3y)/4)^2}}} = 4.5y² + 45y
 {{{((150-3y)^2)/16}}} = 4.5y² + 45y
Multiply through by 16
   (150-3y)² = 72y² + 720y
  [3(50-y)]² = 72y² + 720y 
    9(50-y)² = 72y² + 720y
Divide both sides by 9
     (50-y)² = 8y² + 80y
2500-100y+y² = 8y² + 80y
           0 = 7y² + 180y - 2500
Factoring:
           0 = (y-10)(7y+250)
Use zero-factor property:
               y-10 = 0;  7y+250 = 0
                  y = 10      7y = -250
                               y = {{{-250/7}}}

Discard the negative value.  Substitute y = 10 into

 {{{(150-3y)/4}}} = x

 {{{(150-3(10))/4}}} = x

 {{{(150-30)/4}}} = x

 {{{120/4}}} = x
 
 30 = x

Edwin</pre>