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
![R(n) = 586n](https://tex.z-dn.net/?f=R%28n%29%20%3D%20586n)
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
.
So we have the following piecewise function:
![R(n) = \left \{ {{586n}, n\leq 80 \atop {-5n^{2} + 986n}, n > 80} \right](https://tex.z-dn.net/?f=R%28n%29%20%3D%20%5Cleft%20%5C%7B%20%7B%7B586n%7D%2C%20n%5Cleq%2080%20%5Catop%20%7B-5n%5E%7B2%7D%20%2B%20986n%7D%2C%20n%20%3E%2080%7D%20%5Cright)
The maxium value of a quadratic function in the format of
happens at:
![n_{v} = -\frac{b}{2a}](https://tex.z-dn.net/?f=n_%7Bv%7D%20%3D%20-%5Cfrac%7Bb%7D%7B2a%7D)
The maximum value is:
![y(n_{v})](https://tex.z-dn.net/?f=y%28n_%7Bv%7D%29)
So:
(a) Find the number of passengers that will maximize the revenue received from the flight.
We have to see if
is higher than 80.
We have that, for
,
, so ![a = -5, b = 986](https://tex.z-dn.net/?f=a%20%3D%20-5%2C%20b%20%3D%20986)
The number of passengers that will maximize the revenue received from the flight is:
![n_{v} = -\frac{b}{2a} = -\frac{986}{2(-5)} = 98.6](https://tex.z-dn.net/?f=n_%7Bv%7D%20%3D%20-%5Cfrac%7Bb%7D%7B2a%7D%20%3D%20-%5Cfrac%7B986%7D%7B2%28-5%29%7D%20%3D%2098.6)
Rounding up, the number of passengers that will maximize the revenue received from the flight is 99.
(b) Find the maximum revenue.
This is
.
![R(n) = -5n^{2} + 986n](https://tex.z-dn.net/?f=R%28n%29%20%3D%20-5n%5E%7B2%7D%20%2B%20986n)
![R(99) = -5*(99)^{2} + 986*(99) = 48609](https://tex.z-dn.net/?f=R%2899%29%20%3D%20-5%2A%2899%29%5E%7B2%7D%20%2B%20986%2A%2899%29%20%3D%2048609)
The maximum revenue is $48,609.