SOLUTION: A rancher wishes to enclose a rectangular corral with 360 feet of fencing. The fencing is only required on three sides of an existing stone wall. What are the dimensions of the cor

Algebra ->  Rational-functions -> SOLUTION: A rancher wishes to enclose a rectangular corral with 360 feet of fencing. The fencing is only required on three sides of an existing stone wall. What are the dimensions of the cor      Log On


   

Question 1011756: A rancher wishes to enclose a rectangular corral with 360 feet of fencing. The fencing is only required on three sides of an existing stone wall. What are the dimensions of the corral of maximum area? Use x for the short widths, and y for the longer length.
1. Find equation with x, y, and 360 in it
2. Area as a function of x and y
3. Area as a function of x only
4. Find the dimensions
I would appreciate it if I could see the steps and process, thank you!
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +y+ = the length of fencing parallel
to the existing stone wall
Let +x+ = the length of each of the
other 2 opposite sides
---------------------
+y+%2B+2x+=+360+
+y+=+360+-+2x+
----------------------
Let +A+ = the area of the corral
+A+=+x%2Ay+
+A+=+x%2A%28+360+-+2x+%29+
+A+=+-2x%5E2+%2B+360x+
This is a parabola, and it has a maximum, not a minimum
because of the minus sign with the +x%5E2+ tertm
---------------------------
The formula for the x-coordinate of the vertex is:
+x%5Bmax%5D+=+-b%2F%282a%29+
+x%5Bmax%5D+=+%28-360%29%2F%282%2A%28-2%29%29+
+x%5Bmax%5D+=+360%2F4+
+x%5Bmax%5D+=+90+ ft
------------------
And
+y+=+360+-+2x+
+y+=+360+-+2%2A90+
+y+=+360+-+180+
+y%5Bmax%5D+=+180+ ft
---------------------
The maximum area has dimensions
90 x 180
and
+A%5Bmax%5D+=+90%2A180+=+16200+ ft2
-------------------------------
You can check this. Suppose you add
1 foot to each of the x sides. That means
you must subtract 2 from the y side.
You end up with 89x182
+A+=+89%2A182+=+16198+ ft2
The area is less, as it should be
---------------------------
Try adding 1 ft to each x-side and
subtracting 2 ft from the y-side
Hope this helps