Question 174569
<font size = 7 color = "red"><b>Edwin's solution:</b></font>

A rectangular enclosure is made with 100 ft. of fencing on three sides. The fourth side is the wall of a barn. Find the greatest possible area of such an enclosure. My choices are as follows: A) 400ft^2 B) 625 ft^2 C) 1111.1 ft ^2 D) 2500ft^2.
<pre><font size = 4 color = "indigo"><b>
{{{drawing(200,100,-40,40,-30,10,

line(-40,0,40,0), rectangle(-25,-25,25,0),
locate(-7,7,BARN),locate(-30,-9.5,x), locate(26,-9.5,x),
locate(-10,-27,100-2x) 

 )}}}

{{{matrix(1,5, Area, "=", length, times, width)}}} 

Let y = the area.

{{{matrix(27,4, "",y, "=", (100-2x)x,
               "","","","",
               "",y, "=", 100x-2x^2,
               "","","","",
rearrange_in_descending_order,"","","",
               "","","","",
                "",y, "=", -2x^2+100x,
               "","","","",
factor_-2_out_on_the_right,"","","",
               "","","","",
                "",y, "=", -2(x^2-50x),
take_half_of_50_get_25,"","","",
square_it_getting_625,"","","",
add_and_subtract_625,"","","",
inside_the_parentheses,"","","",
                "","","","",
                "",y, "=", -2(x^2-50x+625-625),
factor_the_first_3_terms,"","","",
in_the_parentheses,"","","",
                "","","","",
                "",y,"=",-2((x-10)(x-10)-625),
                "","","","",
                "",y,"=",-2((x-10)^2-625),
distribute_the_-2_leaving,"","","",
(x+10)^2_intact,"","","",               
                "","","","",
                "",y,"=",-2(x-10)^2 +1250
 )}}}

The vertex of the parabola {{{matrix(1,3,y,"=",a(x-h)^2+k)}}}
is {{{matrix(1,1, "(h,k)")}}}, so the vertex here is {{{matrix(1,1, "(10,1250)")}}},  

The graph is {{{graph(200,200,-10,52,-200,1275,-2(x-10)^2+1250)}}}

So the maximum area is {{{1250ft^2}}}.  That is not one of the
choices, so the problem is botched somehow.

Edwin</pre>