SOLUTION: What is the maximum amount of fencing needed to construct a rectangle enclosure containing 1800 ft^2 using a river as a natural boundary on one side?

Algebra ->  Surface-area -> SOLUTION: What is the maximum amount of fencing needed to construct a rectangle enclosure containing 1800 ft^2 using a river as a natural boundary on one side?       Log On


   

Question 1146865: What is the maximum amount of fencing needed to construct a rectangle enclosure containing 1800 ft^2 using a river as a natural boundary on one side?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Dimensions x and y;
system%28xy=1800%2Cx%2B2y=D%29


y=1800%2Fx%5E2------------mistake here-----the rest needs to be fixed-------


--------------NO-----------based upon above mistake--------------------------
x%2B2%281800%2Fx%5E2%29=D
x%2B3600%2Fx%5E2=D
dD%2Fdx=1%2B3600%2Ax%5E%28-3%29%28-2%29
dD%2Fdx=1-7200%2Fx%5E3
dD%2Fdx=%28x%5E3-7200%29%2Fx%5E3

Find x value for x%5E3=7200.



36*2*10*10
2*3*2*3*2*2*2*5*5
(2^5*3*5^2)
(2^3*2^2)(75)
x=2%2Aroot%283%2C300%29
-
y=1800%2F%282%2Aroot%283%2C300%29%29


If no mistakes made, the total length of fencing:
2%2Aroot%283%2C300%29%2B1800%2Froot%283%2C300%29

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

            Regarding this post,  I have two notices.


1.   Your formulation is  INCORRECT.

      The question should ask about the  MINIMUM  length of the fencing ---- NOT about the maximum length.
      The maximum length  DOES  NOT  EXIST.   You can make your enclosure longer and narrower,  by keeping the same area.

      The correct formulation is  THIS : 

  
           What is the highlight%28minimum%29 fencing length needed to construct a rectangle enclosure 
           containing 1800 ft^2 using a river as a natural boundary on one side?


2.   The "solution" by @josgarithmetic is   TOTALLY   WRONG,  starting from its third line to the end.

      So you better simply  IGNORE  it.


      Below find my correct solution. 


xy = 1800              (1)

x + 2y -----> minimize        (x is the length along the river)



So your task is to minimize (x+2y) under the given condition/restriction  (1).



From (1),  x = 1800%2Fy,  so we need to minimize the function  f(y) = 1800%2Fy+%2B+2y.


The derivative  f'(y) = -1800%2Fy%5E2 + 2.


To find the minimum of f(y), equate its derivative to zero


    -1800%2Fy%5E2 + 2 = 0


    1800%2Fy%5E2 = 2

    y%5E2 = 1800%2F2 = 900

    y = sqrt%28900%29 = 30.


ANSWER.  The minimum fencing is at y = 30 ft perpendicular to the river and x = 1800%2Fy = 1800%2F30 = 60 ft along the river.

Solved.