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19 October, 03:01

A rectangular parking area measuring 5000 ft squared is to be enclosed on three sides using chain-link fencing that costs $5.50 per foot. The fourth side will be a wooden fence that costs $7 per foot. What dimensions will minimize the total cost to enclose this area, and what is the minimum cost?

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  1. 19 October, 04:05
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    Dimensions: 75.3778 ft and 66.3325 ft

    Minimum price: $1658.31

    Step-by-step explanation:

    Let's call the length of the parking area 'x', and the width 'y'.

    Then, we can write the following equations:

    -> Area of the park:

    x * y = 5000

    -> Price of the fences:

    P = 2*x*5.5 + y*5.5 + y*7

    P = 11*x + 12.5*y

    From the first equation, we have that y = 5000/x

    Using this value in the equation for P, we have:

    P = 11*x + 12.5*5000/x = 11*x + 62500/x

    To find the minimum of this function, we need to take its derivative and then make it equal to zero:

    dP/dx = 11 - 62500/x^2 = 0

    x^2 = 65000/11

    x = 250/sqrt (11) = 75.3778 ft

    This is the x value that gives the minimum cost.

    Now, finding y and P, we have:

    x*y = 5000

    y = 5000/75.3778 = 66.3325

    P = 11*x + 62500/x = $1658.31
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