Maximizing Revenue

[turophile]

Here's the problem:

An airline offers a charter flight at a fare of $100 per person if 50 to 100 passengers sign up. For each passenger beyond 100, the fare for each passenger is reduced by fifty cents. The plane has 200 seats. How many passengers will provide the largest total revenue?

I'm having difficulty coming up with the equation that describes the revenue. Any hints?

 

[Deleted member 4993]

Here's the problem:

An airline offers a charter flight at a fare of $100 per person if 50 to 100 passengers sign up. For each passenger beyond 100, the fare for each passenger is reduced by fifty cents. The plane has 200 seats. How many passengers will provide the largest total revenue?

I'm having difficulty coming up with the equation that describes the revenue. Any hints?

First of all let us understand that the number of passenger for maximum revenue is > 100

So let the # of passenger = 100 + P.................... 0 < P ? 100

for P = 1, R = 100*(100+1) - 1*0.5*(100+1)

for P = 2, R = 100*(100+2) - 2*0.5*(100+2)

for P = P, R = 100*(100+P)- P*0.5*(100+P)

 

[soroban]

Hello, turophile!

An airline offers a charter flight at a fare of $100 per person if 50 to 100 passengers sign up.
For each passenger beyond 100, the fare for each passenger is reduced by fifty cents.
The plane has 200 seats.
How many passengers will provide the largest total revenue?

\(\displaystyle \text{Let }x\text{ = number of passengers in excess of 100.}\)

\(\displaystyle \text{Then there are: }\,N \:=\: 100+x \text{ passengers.}\)

\(\displaystyle \text{And each pays a fare of: }\,F\:=\:100 - \tfrac{1}{2}x \text{ dollars.}\)

\(\displaystyle \text{Hence: }\:R \;=\;N\cdot F \:=\100+x)(100-\tfrac{1}{2}x) \:=\:10,\!000 + 50x - \tfrac{1}{2}x^2\)


\(\displaystyle \text{We must maximize }R.\)

.\(\displaystyle R' \,=\,0 \quad\Rightarrow\quad 50 - x \;=\;0 \quad\Rightarrow\quad x \,=\,50\)


\(\displaystyle \text{Therefore, }\:100+50 \:=\:150\text{ passengers will generate maximum revenue.}\)

. . And 150 is within the seating capacity of the plane.