Respuesta :
Answer:
a) The number of passengers that will maximize the revenue received from the flight is 99.
b) The maximum revenue is $48,609.
Step-by-step explanation:
We have to analyse two cases to build a piecewise function.
If there are 80 or less passengers, we have that:
The cost of the trip is $586 for each passenger. So
[tex]R(n) = 586n[/tex]
If there are more than 80 passengers.
There is a refund of $5 per passenger for each passenger in excess of 80. So the cost for each passenger is
[tex]R(n) = (586 - 5(n-80))n = -5n^{2} +400n + 586n = -5n^{2} + 986n[/tex].
So we have the following piecewise function:
[tex]R(n) = \left \{ {{586n}, n\leq 80 \atop {-5n^{2} + 986n}, n > 80} \right[/tex]
The maxium value of a quadratic function in the format of [tex]y(n) = an^{2} + bn + c[/tex] happens at:
[tex]n_{v} = -\frac{b}{2a}[/tex]
The maximum value is:
[tex]y(n_{v})[/tex]
So:
(a) Find the number of passengers that will maximize the revenue received from the flight.
We have to see if [tex]n_{v}[/tex] is higher than 80.
We have that, for [tex]n > 80[/tex], [tex]R(n) = -5n^{2} + 986n[/tex], so [tex]a = -5, b = 986[/tex]
The number of passengers that will maximize the revenue received from the flight is:
[tex]n_{v} = -\frac{b}{2a} = -\frac{986}{2(-5)} = 98.6[/tex]
Rounding up, the number of passengers that will maximize the revenue received from the flight is 99.
(b) Find the maximum revenue.
This is [tex]R(99)[/tex].
[tex]R(n) = -5n^{2} + 986n[/tex]
[tex]R(99) = -5*(99)^{2} + 986*(99) = 48609[/tex]
The maximum revenue is $48,609.
