For a particular flight from dulles to sf, an airline uses wide-body jets with a capacity of 440 passengers. it costs the airline $4,000 plus $70 per passenger to operate each flight. through experience the airline has discovered that if a ticket price is $t, then they can expect (440−0.64t) passengers to book the flight. determine the ticket price, t, that will maximize the airline's profit.

$378.75

Explanation:

Data provided:

Capacity = 440 passengers

Operating cost = $4,000 + $70(Number of passengers)

Expected number of passengers = 440 - 0.64T

Ticket price = T

Total operational cost = $4000 + $70( 440-0.64T )

Total operational cost = $34,800 - 44.8T

Thus,

Total revenue = Number of passengers × Ticket price

= (440 - 0.64T)T

= 440T - 0.64T²

also,

Total profit ,P(T) = Total revenue - Total operational cost

P(T) = ( 440T - 0.64T²) - (34,800 - 44.8T)

P(T) = - 0.64T² - 34,800 + 484.8T

Now,

Differentiating with respect to ticket price T

P'(T) = -0.64(2)T - 0 + 484.8(1)

or

P'(T) = - 1.28T + 484.8 (1)

For point of maxima or minima

P'(T) = 0

or

 - 1.28T + 484.8 = 0

or

1.28T = 484.8

or

T = $378.75

now,

again differentiating (1) to check for maxima or minima

P''(T)= -1.26(1) + 0

P''(T) = -1.26

Since,

P"(T)  < 0

Hence,

T = $378.75 will maximise the profit