Question 1099767: a) A rectangular pen is build with one side against a barn. If 2100 m of fencing are used for the other three sides of the pen, what dimensions maximize the area of the pen?
b) A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 400m^2. What are the dimensions of each pen that minimize the amount of fence that must be used?
Found 3 solutions by josmiceli, KMST, ikleyn:
Answer by josmiceli(19441)   (Show Source):
Answer by KMST(5328)  (Show Source):
You can put this solution on YOUR website! I see the first question as a quadratic function question,
although you need to know enough geometry to understand what a rectangle is.
The second question, looks more like a calculus question
a)  The area  of that pen (in square meters) is  .
You should recognize that as a quadratic function,
which graphs as a parabola, looking like this:
 , with two zeros and a maximum exactly midway between them.
Re-writing it as  shows you clearly that
 for  and  ,
 in between, for  ,
and the maximum is at 
The dimensions that maximize the area are
 for the length of each of the two fencing sides attached to the barn wall, and
 for the length of of the fencing side parallel to the barn wall.
b) If we design something with four identical pens, like this:
 , where  and are lengths in m,
we know that  <--->  ,
and that makes the total length of fence needed, , in m
 .
Maybe you are supposed to use a graphing calculator to find that the minimum for happens at approximately  ,
and you could tell the farmer to use 5  length of fencing perpendicular to the barn wall,
attached to a   length parallel to the wall.
Using calculus, you would find that the derivative is
 .
As  for  ,
 for  , and  for  ,
the function decreases for  to a minimum at   , and increases for .
If there is another alternative approach, let me know.
Answer by ikleyn(52798)  (Show Source):
|
|
|
