SOLUTION: A rectangular field beside a river is to be fenced by 80 meters of fencing. No fence is needed along the riverbank. What are the dimensions of the field that maximize its area?

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Question 1157514: A rectangular field beside a river is to be fenced by 80 meters of fencing. No fence is needed along the riverbank. What are the dimensions of the field that maximize its area?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
x, x, 80-2x
The three lengths
80-2x is length parallel to riverbank

x%2880-2x%29, area

Maximum area should be in exact middle between the zeros.
One zero is 0.
Other zero is 80-2x=0
40-x=0
x=40

Exact middle of 0 and 40 is 20.

Dimensions to max area:
20 and 40.

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

Since one side is the river, the rectangle's fence perimeter will be
L + 2W = 80.

Hence, L = 80 - 2W.

Area = Length * Width.

Substitute (80-2W) for L:

    A = W(80 - 2W)

    A = -2W^2 + 80W.

This is a quadratic function. It has the maximum at x = -b/(2a), according to the general theory.

    (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

     in this site).


For our quadratic function the maximum is at

    W = -80%2F%282%2A%28-2%29%29 = %28-80%29%2F%28-4%29 = 20.

So, W = 20 meters is the width for max area.

Then the length is  L = 80 - 2W = 80 - 2*20 = 40 meters

Find the max area. Substitute 20 for W

    A = -2(20^2) + 80*20 = 800 square meters.

The plot of the quadratic function for the area is shown below:  y = area and x = width.

+graph%28+300%2C+200%2C+-50%2C+50%2C+-100%2C+1000%2C+-2x%5E2+%2B+80x%29+


My other lessons in this site on finding the maximum/minimum of a quadratic function are
    - 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
    - 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 rancher planning to fence two adjacent rectangular corrals to enclose the maximal area
    - Finding the maximum area of the window of a special form
    - Using quadratic functions to solve problems on maximizing revenue/profit
    - Minimal distance between sailing ships in a sea
    - Advanced lesson on finding minima of (x+1)(x+2)(x+3)(x+4)
    - OVERVIEW of lessons on finding the maximum/minimum of a quadratic function

Use this file/link  ALGEBRA-I - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-I.