Question in the description

Question

Optimal Chapter-Flight Fare

 If exactly 208 people sign up for a charter flight, Leisure World Travel Agency charges $310/person. However, if more than 208 people sign up for the flight (assume this is the case), then each fare is reduced by $1 for each additional person. Determine how many passengers will result in a maximum revenue for the travel agency. Hint: Let x denote the number of passengers above 208. Show that the revenue function R is given by R(x) = (208 + x)(310 − x).

 

 passengers

What is the maximum revenue?

$  

What would be the fare per passenger in this case?

   dollars per passenger

Answer 1

Let x be the number of passengers above 208.

Then:

Number of passengers = 208 + x

Fare per passenger = 310 - x

So the revenue function is:

R(x) = (208 + x)(310 - x)

Expand:

R(x) = 64,480 + 102x - x²

Since this is a downward-opening parabola, maximum revenue occurs at its vertex:

x = -b/(2a)

x = -102/[2(-1)]

x = 51

Therefore, the total number of passengers is:

208 + 51 = 259 passengers

Now find the maximum revenue:

R(51) = (259)(310 - 51)

= (259)(259)

= $67,081

The fare per passenger is:

310 - 51 = $259

Final answers:

Passengers: 259

Maximum revenue: $67,081

Fare per passenger: $259