SOLUTION: I am a freshman in honors algebra 2 and we're working on applying prabolas to real world senarios. The question is " find the demensions and maximum area of a rectangle with a peri

                 A = L×W  where A = area, L = length and W = wifth
                 P = 2L + 2W where P = perimeter
                 
Let the length be x

Since the perimeter is 48, we substitute in

                 P = 2L + 2W
                48 = 2x + 2W
Divide every term by 2
                24 = x + W
Solve for W

           24 - x  = W

Let the area be y.  Substitute in

                 A = L×W

                 y = x(24 - x)

That equation has this graph: 



Since y = the area, the area will be a maximum at the
peak point, known as "the vertex".

There are three ways to find the vertex.  I don't know which way
your teacher will require you to use.

Method 1 for finding the vertex:

Find the two x-intercepts

 y = x(24 - x)
 
     x(24 - x) = 0
    x=0,  24-x = 0
            -x = -24
             x = 24

The x-intercepts are (0,0) and (24,0)
The vertex occurs halfway when the value of x is halfway
between 0 and 24, or at x=12.

Substituting x=12

 y = 12(24-12)
 y = 12(12)
 y = 144

So the vertex of that parabola is (12,144)

That means the area y will have a maximium area of 144 square units
when the length is x = 12 inches.  That is, when the rectangle is a
12in × 12in square. 
 
-----------------------------------------

Method 2 for finding the vertex:

 y = x(24 - x)

Put the equation in the vertex form:

                 y = a(x-h)² + k

 y = x(24 - x)
 y = 24x - x²
 y = -x² + 24x
Factor out the coefficient of x², which is -1
 y = -1[x² - 24x      ]

Complete the square by multiplying the coefficient of x by 
ans squaring:  -24() = -12, Squaring (-12)² = 144
Add and subtract 144 inside the parentheses:

 y = -1[x² - 24x + 144 - 144]

Factor the first three terms in the brackets as the square of a
binomial

 y = -1[(x - 12)² - 144]

Remove the bracket leving the paretheses intact:

y = -1(x - 12)² + 144

Compare to

y = a(x-h)² + k

Vertex = (h,k) = (12,144)

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

Method 3 for finding the vertex:

Use the vertex formula:

The vertex is (h,k) where

h =  and k = 

 y = -x² + 24x

 y = -x² + 24x + 0

a = -1, b = 24, c = 0

h =  and k = 
h =  = -12
k = 
k = -1(12)^2+24(12)+0
k = -(144)+ 288
k = 144

So the vertex is (12,144)

Use whichever method for finding the vertex your teacher
expects you to use, not necessarily the easiest way.

Edwin