Calculus

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A rectangular garden of area 75 sq ft is to be surrounded on 3 sides by a brick wall costing $12 per foot and on one side by a fence costing $7 per foot. Find the dimensions of the garden that the cost of the materials is minimized Read more: A rectangular garden of area 75 sq ft is to be surrounded on 3 sides by a brick wall costing $10 per foot and on one side by a fence costing $5 per foot. Find the dimensions of the garden that the cost of the materials is minimized

  • Calculus -

    Let the dimensions of the garden be
    x (fence side) and 75/x (since the area is 75).
    So the total cost, C(x)
    = 7x+2*12(75/x)+12(7)
    = 7x+1800/x+84
    To find the minimum cost (at the expense of the shape of the garden)
    we calculate
    C'(x)=0
    7-1800/x^2=0
    x=sqrt(1800/7)=16.04'
    75/x=4.68'

  • Calculus -

    the answer is wrong

  • Calculus -

    let the side with the fence be x ft long, then the perpendicular side is 75/x

    cost = 7x + 12(75/x)
    d(cost)/dx = 7 - 900/x^2 = 0 for a min of cost
    7 = 900/x^2
    x^2 = 900/7
    x = 30/√7 = appr 11.339

    so the garden has to be 11.339 by 75/11.339
    or 11.34 by 6.61

    check: 11.34 x 6.61 = 74.96 , not bad
    cost = 7(11.339) + 900/11.339 = 158.75 --- minimum cost

    let x = 11 , cost = 7(11) + 900/11 = 158.82 , a higher cost
    let x = 11.6, cost = 7(11.5) + 900/11.5 = 158.75 -- just a bit higher
    MY answer is correct

  • Calculus -

    You're right, I mis-calculated the cost of the fourth side (12(7) instead of 12x).

    Let the dimensions of the garden be
    x (fence side) and 75/x (since the area is 75).

    Since there are four sides to the garden:

    So the total cost, C(x)
    = 7x+2*12(75/x)+12(x)
    = 19x+1800/x
    To find the minimum cost (at the expense of the shape of the garden)
    we calculate
    C'(x)=0
    19-1800/x^2=0
    x=sqrt(1800/19)=9.733' (fence side)
    75/x=7.706' (wall side)
    C(sqrt(1800/19))=$369.86

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