What rate should they charge to maximize their revenue? What is their maximum revenue at this rate?

Question

A hotel finds that if they charge $300 per room, they can book 100 rooms. They also find that for every $50 increase, they lose 5 bookings. What rate should they charge to maximize their revenue? What is their maximum revenue at this rate?

Answer 1

First, the relationship between B, bookings and R, rate is linear

 

the slope of that relationship is (100-95)/(300-350)=-1/10

 

The equation for the relationship is -1/10 = (B-95)/(R-350) => B=(1300-R)/10

 

Revenue = BR = (1300R - R²)/10

 

The maximum occurs where (1300 - 2R) = 0 => R = 650

 

At that rate the hotel will book 65 guests and revenue will be 42900