Eric T asked in 科學及數學數學 · 8 years ago

2 calculus questions

1. A parcel delivery service will deliver a package only if the lengthplus the girth (distance around, taken perpendicular to the length)does not exceed 104 inches. Find the maximum volume of a rectangularbox with square ends that satisfies the delivery company'srequirements.

2. For the cost function

圖片參考:http://r140.n100.queensu.ca:8081/wwtmp/equations/5...

find

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(a) The production level that will minimize the average cost.

Ans: 62.996

(b) The minimal average cost

Ans: 53.811

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    1) Let x inches be a side of the square end, then we have:

    Length L <= 104 - 4x

    So taking max. possible volume, we let L = 104 - 4x, then vol. is:

    V = x2(104 - 4x)

    = 104x2 - 4x3

    dV/dx = 208x - 12x2

    d2V/dx2 = 208 - 24x

    When dV/dx = 0,

    208x - 12x2 = 0

    4x(52 - 3x) = 0

    x = 0 (rej.) or 52/3

    When x = 52/3, d2V/dx2 = 208 - 24 x 52/3 < 0

    Hence x = 52/3 gives max. V = 10415 cubic inches

    2a) The avg. cost is:

    A = C/x = 1000/x + 30 + 0.002x2

    dA/dx = -1000/x2 + 0.004x

    d2A/dx2 = 2000/x3 + 0.004

    For dA/dx = 0:

    1000/x2 = 0.004x

    x3 = 250000

    x = 62.996

    Sub into d2A/dx2 = -0.004 > 0

    Hence it gives a min. average cost

    b) Min. avg cost = 53.811 when sub x = 62.996 into C/x

    2011-11-26 13:30:50 補充:

    For 2a) d^2A/dx^2 = 0.012 > 0 when x = 62.996

    Source(s): 原創答案
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