Rental income. A 300-room hotel in Las Vegas is filled to capacity every night at $80 a room. For...

Question

Rental income. A 300-room hotel in Las Vegas is filled to capacity every night at $80 a room. For each $1 increase in rent, 3 fewer rooms are rented. If each rented room costs $10 to service per day, how much should the management charge for each room to maximize gross profit? What is the maximum gross profit?

Robert · Accepted Answer

let​ ​x​ ​represent​ ​the​ ​number​ ​of​ ​$1​ ​increase​ ​in​ ​room​ ​price 
==>​ ​room​ ​price​ ​=​ ​80​ ​+​ ​x(1)​ ​=​ ​80​ ​+​ ​x 
==>​ ​rooms​ ​filled​ ​=​ ​300​ ​-​ ​3x 
Revenue​ ​=​ ​price​ ​per​ ​room​ ​*​ ​number​ ​of​ ​rooms 
==>​ ​Revenue​ ​=​ ​(80​ ​+x)(300​ ​-3x)​ ​=​ ​24000​ ​-240x​ ​+​ ​300x​ ​-3x​2 
==>​ ​-3x​2​​ ​+​ ​60x​ ​+​ ​24000 
total​ ​service​ ​cost​ ​=​ ​number​ ​of​ ​rooms*​ ​cost​ ​per​ ​room​ ​=​ ​(300​ ​-3x)(10)​ ​=​ ​3000​ ​-30x 
Profit​ ​P(x)​ ​=​ ​Revenue​ ​-​ ​Cost​ ​=​ ​(-3x​2​​ ​+​ ​60x​ ​+​ ​24000)​ ​-​ ​(3000​ ​-​ ​30x) 
==>​ ​Profit​ ​P(x)​ ​=​ ​-3x​2​​ ​+​ ​90x​ ​+​ ​21000 
Maximum​ ​profit​ ​==>​ ​P​ ​'(x)​ ​=​ ​0 
==>​ ​-6x​ ​+​ ​90​ ​=​ ​0 
==>​ ​x​ ​=​ ​90/6​ ​=​ ​15 
Management​ ​should​ ​charge​ ​$15 
==>​ ​Maximum​ ​profit​ ​=​ ​-3(15)​2​​ ​+​ ​90(15)​ ​+​ ​21000​ ​=​ ​21675$

Rental income. A 300-room hotel in Las Vegas is filled to capacity every night at $80 a room. For each $1 increase in rent, 3 fewer rooms are rented. If each rented room costs $10 to service per day,...

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