I think the text I’m using to teach myself calculus (“Forgotten Calculus” by Barbara Lee Bleau) might be incorrectly setting up an optimization problem.
I will state the problem, show her way of setting it up, and then show mine. I will not consider the actual solution, as I am clear on what to do after the setup.
I’d welcome comment on who is right. If I’m wrong, guidance as to why would be very helpful.
A tour agency has signed up 100 people for a cruise on a ship with a maximum capacity of 150.
A ticket costs $2000. For each $5 that this price is lowered, one new passenger signs up.
Incidental costs to the agency are $500 per passenger. There are also fixed costs of $125,000 for the ship.
By how much should the price be lowered to maximize the agency’s profit?
Author’s setup of the total profit formula
When x (apparently; she doesn’t state it explicitly) equals the number of passengers in excess of the original 100, and 0 <= x <= 50,
Profit = Revenue – variable costs – fixed costs
= (price * quantity) – variable costs – fixed cost
= [ (2,000 – 5x)(100 + x) ] – 500x – 125,000
My setup of the total profit formula is the same as hers, except in the definition of variable costs.
I’d define variable costs as [500 (100 +x) ], versus her definition of 500x.
My definition multiplies the variable cost of 500 by the total number of passengers; hers seems to multiply the variable cost of 500 by only the number of passengers in excess of 100.
I find this counterintuitive. Why would the addition of any passengers over the original 100 suddenly render the original 100 “costless”?