Question

A 400​-room hotel can rent every one of its rooms at ​$50 per room. For each​ $1 increase in​ rent, 4 fewer rooms are rented. Each rented room costs the hotel​ $10 to service per day. How much should the hotel charge for each room to maximize its daily​ profit? What is the maximum daily​ profit?

Answer 1

The answer is: The hotel should charge $75 per room to earn a maximum daily profit of $16,900

Step-by-step explanation:

If the hotel rents its 400 rooms at $50, its profit will be:

(400 x $50) - ($400 x $10) = $20,000 - $4,000 = $16,000

Since the price elasticity of the demand is inelastic (PES = 0.5, a 2% increase in the price will only decrease the quantity demanded in 1%), the hotel should increase its price.

let P = number of $1 increases in the price

max. profit = ($40 - $10 + P) x (400 - 4P) = ($30 + P) x (400 - 4P) =

max. profit = -4P² + 280P + $12,000

The vertex (maximum point) of a parabola is given by P = -b/(2a)

• where b= 280 ; a = -4

P = -280 / -8 = 35

So the price of the hotel rooms should increase by $35 to $75.

Now we substitute P = 35

(-4 x 35²) + (280 x 35) + 12,000 = -4,900 + 9,800 + 12,000 = $16,900