SOLUTION: 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
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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): You can put this solution on YOUR website!
(a)
Let = the length of the
side perpendicular to the barn
= the length of the side
parallel to the barn
-------------------------------
The formula for the W-value of the
maximum is:
and
---------------------------------
The dimensions that maximize area are:
525 x 1050
-----------------------
check:
m2
and
m2
OK
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(52799) (Show Source): You can put this solution on YOUR website!
.
For purely algebraic approach to solve such problems see the lessons
- 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.
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".
Save the link to this online textbook together with its description
Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson
to your archive and use it when it is needed.
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