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1 December, 09:22

A rectangular parking area measuring 6000 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 $6 per foot. What dimensions will minimize the total cost to enclose this area, and what is the minimum cost?

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  1. 1 December, 12:45
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    length x = 79.2 ft

    width y = 75.75 ft

    minimum cost P = Rs 1742.41212

    Step-by-step explanation:

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

    Then, we can write the following equations:

    Then, Area of the park: A = x*y = 6000

    Now, Price of the fences P = 2*5.5x + 5.5y + 6y

    P = 11x + 11.5y

    From the first equation, we have that y = 6000/x

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

    P = 11x + 11.5*6000/x = 11x + 69000/x

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

    ⇒ dP/dx = 11 - 69000/x^2 = 0

    ⇒x^2 = 69000/11

    x = 79.20 ft

    This is the x value that gives the minimum cost.

    Now, finding y and P, we have:

    x*y = 6000

    y = 6000/79.2 = 75.75 ft

    P = 11x + 69000/x = Rs 1742.41212
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