# 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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