SOLUTION: Boris has 320 meters of fencing. He will use it to form three sides of a rectangular garden. The fourth side will be along a house and will not need fencing. (a) Find a function

Question 1189543: Boris has 320 meters of fencing. He will use it to form three sides of a rectangular garden. The fourth side will be along a house and will not need fencing.
(a) Find a function that gives the area A (x) of the garden (in square meters) in
terms of x.
A(x) =?
(b) What side length x gives the maximum area that the garden can have?
Side length x : ? meters

(c) What is the maximum area that the garden can have?
Maximum area: ? square meters
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.

            Instead of solving your problem, I'll give you the solution to a TWIN problem,
            to give you an opportunity to learn the method and the subject.

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A farmer plans to fence a rectangular grazing area along a river with 300 yards of fence.
What is the largest area he can enclose?

Solution

Since one side is the river, the rectangle's fence perimeter will be

    L + 2W = 300.

Hence, L = 300 - 2W.


    Area = Length * Width.


Substitute (300-2W) for L:

    A = W(300 - 2W)

    A = -2W^2 + 300W.


It 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 =  =  = 75.


So, W = 75 yards is the width for max area.


Then the length is  L = 300 - 2W = 300 - 2*75 = 150 yards.


Find the maximum area. It is

    A = L*W = 150*75 = 11250 square yards.


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

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

    - OVERVIEW of lessons on finding the maximum/minimum of a quadratic function

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