SOLUTION: A rancher has 98 meters of fence with which to enclose three sides of a rectangular plot (the fourth side is a river and will not require fencing). Find the dimensions of the plot

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Question 1087313: A rancher has 98 meters of fence with which to enclose three sides of a rectangular plot (the fourth side is a river and will not require fencing). Find the dimensions of the plot with the largest possible area. (For the purpose of this problem, the width will be the smaller dimension (needing two sides); the length with be the longer dimension (needing one side).)
Found 2 solutions by josmiceli, ikleyn:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +L+ = the length parallel to the river
Let +W+ = the width of the other 2 sides
Let +A+ = the area of the rectangular plot
---------------------------------------
+98+=+L+%2B+2W+
+L+=+98+-+2W+
+A+=+L%2AW+
+A+=+%28+98+-+2W+%29%2AW+
+A+=+-2W%5E2+%2B+98W+
-----------------------
The formula for +W%5Bmax%5D+ is:
+W%5Bmax%5D+=+-b%2F%28+2a+%29+
+a+=+-2
+b+=+98+
+W%5Bmax%5D+=+-98%2F%28+2%2A%28-2%29+%29+
+W%5Bmax%5D+=+24.5+
Put this value back into equation to get +A%5Bmax%5D+
+A%5Bmax%5D+=+-2%2A24.5%5E2+%2B+98%2A24.5+
+A%5Bmax%5D+=+-1200.5+%2B+2401+
+A%5Bmax%5D+=+1200.5+ m2
The maximum area is 1200.5 m2
---------------------------------
Here's the plot:
+graph%28+400%2C+400%2C+-5%2C+50%2C+-140%2C+1300%2C+-2x%5E2+%2B+98x+%29+
check my math!

Answer by ikleyn(52847) About Me  (Show Source):
You can put this solution on YOUR website!
.
For finding the maximum/minimum of a quadratic function see the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola
in this site.

For finding the maximum area of rectangles at different conditions see the lessons
    - A rectangle with a given perimeter which has the maximal area is a square
    - A farmer planning to fence a rectangular garden to enclose the maximal area
    - A farmer planning to fence a rectangular area along the river to enclose the maximal area (*)
    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area
in this site.

Notice that the lesson marked (*) in this list contains the solution exactly of the same problem as in your post.


Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".