A family wants to build a rectangular garden on one side of a barn. If 600 feet of fencing is available to use, then what is the area of t

Question

A family wants to build a rectangular garden on one side of a barn. If 600 feet of fencing is available to use, then what is the area of the largest garden that could be built?
(A) Define a function that relates the area enclosed by the fence (in 〖ft.〗^2) in terms of its length (in ft.)
(B) What is the practical domain of this function?
(C) What is the largest area that the fence could enclose?

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Nick 1 year 2021-08-30T13:35:34+00:00 1 Answers 0 views 0

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    2021-08-30T13:37:01+00:00

    Answer: (A) A=300l-l^{2}

                   (B) Length varies between 1 and 150

                   (C) Largest area is 22500ft²

    Step-by-step explanation: Suppose length is l and width is w.

    The rectangular garden has perimeter of 600ft, which is mathematically represented as

    2l+2w=600

    Area of a rectangle is calculated as

    A=lw

    Now, we have a system of equations:

    2l+2w=600

    A=lw

    Isolate w, so we have l:

    2w=600-2l

    w = 300 – l

    Substitute in the area equation:

    A = l(300 – l)

    A = 300l – l²

    (A) Function of area in terms of length is given by A = 300l – l²

    (B) The practical domain for this function is values between 1 and 150.

    (C) For the largest area, we need to determine the largest garden possible. For that, we take first derivative of the function:

    A’ = 300 – 2l

    Find the values of l when A’=0:

    300 – 2l = 0

    2l = 300

    l = 150

    Replace l in the equation:

    w = 300 – 150

    w = 150

    Now, calculate the largest area:

    A = 150*150

    A = 22500

    The largest area the fence can enclose is 22500ft².

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