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A farmer has 230 ft of fence to enclose a rectangular garden. What is the largest garden area that can be enclosed with the 230 ft of fence? Explain your work.

  • Math -

    Hi Crystal,

    Good question! I hope this explanation helps. :)

    You know your farmer has 230 feet of fencing, so we're going to keep that.

    You're looking for Area, so what is the area of a rectangle?

    Area = x * y

    But, first you need to define one of your variables in order to proceed. We do this by taking the perimeter of your rectangle. To get perimeter, you add the 4 sides, 2 of the length and 2 of the width.

    Perimeter = 2x + 2y

    Now, remember your 230 feet, that is the total perimeter you can possibly have because it's the maximum amount of fencing you have. Plug that in for P!

    230 = 2x + 2y

    Now, solve for one of your variables. Personally, I almost always solve for y because in a quadratic I prefer to work with x's.

    So:

    230 = 2x + 2y
    -2x -2x

    230 - 2x = 2y
    _____________
    2 (to get y alone)

    115 - x = y

    Great! Now you have defined one of your terms! You have a value for y. Plug that value for y in as y in your area formula, and solve.

    A(x) = x * y
    A(x) = x * (115 - x)
    A(x) = 115x - x^2

    You have a quadratic now:

    A(x)= -x^2 + 115 x

    Now, once you have it in this form, remember the form of a quadratic equation:

    Ax^2 + Bx + C (A, B, and C are just your coefficients and they are integers)

    To find your maximum area, you need to use this formula:

    x = -B Here, B = 115
    _____

    2(A) Here, A = -1

    So you have:

    x = - 115
    ______
    2 (-1)

    x = -115
    ______
    -2

  • Math (Continued) -

    x = -115
    ____
    -2

    Solve this to get: 57.5
    So, your greatest value for x will be 57.5.

    Plug this in to your perimeter equation to determine the value of y.

    Remember:

    P(x) = 230 = 2x + 2y
    230 = 2(57.5) + 2y
    230 = 115 + 2y
    -115 -115
    ___________________
    115 = 2y
    ___ ___
    2 2

    57.5 = y

    You have a sqaure! You now know that the value of x that will produce your maximum area is 57.5 and your value for y that will produce maximum area is 57.5.

    Now, remember your area formula?

    A(x) = x * y

    Plug in your variables to find maximum area :)

    A(x) = 57.5 * 57.5 = 3,306.25 Feet

    Your maximum area = 3,306.25 Feet.

  • Math -

    37694046292365

  • Math -

    P=2x+2y(perimeter)
    JUST TO ADD TO THE ABOVE SOLUTION.
    230=2x+2y
    2y=230-2x
    y=115-x

    A=xy(Area)
    A=x[115-x]
    A=115x-x^2
    dA/dX=115-2X=0(at maximum)
    115=2x
    x=57.5(CRITICAL VALUE)

  • Math -

    105 feet by 105 feet

  • Math -

    57.5

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  • Math -

    Yeaaaaaaa Boiiiiiiiiii

  • Math -

    A farmer wants to know the area of a garden that is 312312 meters by 125125 meters. He uses the given model for help. Which expression can be used to find the area in square meters?

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