Help setting up an optimization problem

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.

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The problem

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?

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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

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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”?

Quote:

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Originally Posted by lingyai

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.

-------------------

The problem

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”?

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It doesn't make any difference to the price that maximises profit, as
the the profit in the two models differ by a fixed amount.

Of course it does make a difference to the profit at the optimal ticket
price.

(there is probably a misunderstanding of what is meant by variable
cost here, where the variable is the number of extra passengers, not
the total number of passenger. Presumably the author has lumped the
costs of the base 100 people in the fixed costs)

RonL