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Calculus/Optimization

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Expert: - 4/12/2010


Question
If 40 passengers hire a special car on a train, they will be charged 8 dollars each, but this fare will be cut by 10 cents apiece for all passengers, for each passenger of the 40; thus, 50 passengers would pay 7 dollars each. What number of passengers will produce the greatest income for the railroad?

Answer
If 40 passengers is $8 each and 50 passengers are $7 each,
that appears that each passengerover cuts the price by 10c each.

This means the total revenue is (x-40)(8 - 0.1x).  That is R = -0.1x² + 12x - 3200.

This means dR/dx = -0.2x + 12.  Maximizing means setting dR/dx = 0.  This gives 0.2x = 12.
Note the 5(0.2) = 1, so the answer is x = 5(12) = 60.

So that is the number of passengers that will maximize the profit.

Calculus

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Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.

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