Calculus

A company offers the following schedule of charges: $30 per thousand for orders of 50,000 or less with the charge per thousand decreased by 37.5 cents for each thousand above 50,000. Find the order which will make the company's receipts a maximum?

How do you solve it?

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  1. right now:
    cost per thousand = 30
    orders = 50 000

    for every increase of 1000 orders, cost decreases by .375

    net the number of 1000 increases be n
    cost per thousand = 30 - .275n
    orders = 50,000 + 1000n

    receipts = (30-.375n)(50000 + 1000n)
    = 1,500,000 + 30,000n - 18750n - 375n^2
    = -375n^2 + 11250n +1500000
    d(receipts)/dn = -750n + 11250
    = 0 for a max of receipts
    750n = 11250
    n = 15

    for a max recepts
    number should be 50000+1000(15) = 65000
    cost per thousand = 30 - .375(15) = $24.375

    testing:
    for the above answer :
    receipts = 65000(24.375) = 1, 584,375

    take n = 14
    number = 64000
    cost = 30-.375(14) = $24.75
    receipts = 64000(24.75) = 1, 584,000 , ahh a bit less

    take n = 16
    number = 66000
    cost = 30-.375(16) = $24
    receipts = 66000(14) = 1,584,000 , again a bit less

    My answer is correct.

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  2. what is the final answer?

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