You own a small airplane that holds a maximum of 20 passengers. It costs you $10

Question

You own a small airplane that holds a maximum of 20 passengers. It costs you $100 per flight from St. Thomas to St. Croix for gas and wages plus an additional $6 per passenger for the extra gas required by the extra weight. The charge per passenger is $30 each if 10 people charter your plane (10 is the minimum number you will fly), and this charge is reduced by $1 per passenger for each passenger over 10 who travels (that is, if 11 fly they each pay $29, if 12 fly they each pay $28, etc.). What number of passengers on a flight will maximize your profit? If possible, please type answer

Explanation / Answer

Let x be the number of $1 reductions in price.

Then revenue = (30 - x)(10 + x) = 300 + 20x - x^2

Cost of flying (10 + x) passengers = 100 + 6(x + 10) = 6x + 160

Profit = Revenue - Cost = 300 + 20x - x^2 - 6x - 160 = 140 + 14x - x^2

For maximum prodit, the derivative = 0

14 - 2x = 0

This gives x = 7

Number of passengers that will maximize the profit = 10 + 7 = 17

Maximum profit = 140 + 14(17) - 17^2 = $89.

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