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If exactly 220 people sign up for a charter flight, Leisure World Travel Agency charges $282/person. However, if more than 220 people sign up for the flight (assume this is the case), then every fare is reduced by $1 times the number of passengers above 220. Determine how many passengers will result in a maximum revenue for the travel agency. Hint: Let x denote the number of passengers above 220. Show that the revenue function R is given by R(x)

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Answer:

251 passengers will result in a maximum revenue.

Step-by-step explanation:

The price per ticket of the first 220 passengers is given by:

[tex]220*(282-x)[/tex]

The price per ticket of the additional x passengers is:

[tex]x(282 - x)[/tex]

Adding both parts gives us the revenue function R(x):

[tex]R = 220*(282 -x) + x(282-x)[/tex]

The term (282-x) is present in both parts and can be factored:

[tex]R(x)= (220+x)*(282-x)\\R(x)= -x^2 +62x+ 62,040[/tex]

To find how many passengers will result in a maximum revenue, derive the function R(x) and find its zeroes:

[tex]\frac{d}{dx}R(x)= \frac{d}{dx} (-x^2 +62x+ 62,040)\\\frac{d}{dx}R(x)=-2x +62 = 0\\x=\frac{62}{2}\\x=31[/tex]

31 passengers above 220 will result in a maximum revenue. Therefore, 251 passengers will result in a maximum revenue.

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