a hotel chain charges $120 per room and rents an avg of 40 rooms per night at this rate. For each $10 reduction in price, they rent an avg of 10 more rooms. Find the rate they should be charging to maximize revenue.

Help please! Thanks!

To find the rate that the hotel chain should be charging to maximize revenue, we can use the concept of elasticity of demand. The elasticity of demand measures how sensitive the demand for a product (in this case, hotel rooms) is to changes in price.

Let's start by considering the effect of a $10 reduction in price on the number of rooms rented. According to the information given, a $10 reduction increases the average number of rooms rented by 10.

Therefore, we can say that the elasticity of demand is = (10 / 40) = 0.25. This means that for every 1% reduction in price, there is a 0.25% increase in the quantity demanded.

Now, let's calculate the demand function for the hotel rooms. We can assume that the demand function follows a linear relationship. Let's denote the price as P and the number of rooms rented as Q.

Demand function: Q = a - bP, where a is the intercept and b is the slope of the demand curve.

In our case, we know that at a price of $120, the average number of rooms rented is 40, so we can substitute these values into the demand function:

40 = a - b(120)

Next, we need another point on the demand curve to solve for both a and b. To get this point, we can use the information that a $10 reduction in price leads to an increase of 10 rooms rented:

40 + 10 = a - b(110)

Simplifying this equation:

50 = a - 110b

We now have a system of two equations:

1) 40 = a - 120b
2) 50 = a - 110b

Solving these equations simultaneously will give us the values of a and b.

Subtracting equation 2 from equation 1 eliminates a:

40 - 50 = -120b + 110b
-10 = -10b

Dividing both sides by -10:

b = 1

We can now substitute the value of b back into either of the equations to solve for a. Let's use equation 1:

40 = a - 120(1)
40 = a - 120
a = 160

Therefore, the demand function is:

Q = 160 - P

To maximize revenue, we need to find the price that maximizes the revenue function. Revenue is calculated by multiplying the price by the quantity:

R = P * Q
R = P(160 - P)
R = 160P - P^2

To find the price that maximizes revenue, we need to find the derivative of the revenue function and set it equal to zero:

dR/dP = 160 - 2P

Setting this equal to zero and solving for P:

160 - 2P = 0
2P = 160
P = 80

Therefore, to maximize revenue, the hotel chain should be charging $80 per room.