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 they should be charging to maximize revenue, we need to analyze the relationship between the price charged and the number of rooms rented.

Let's start by assuming that the original price charged is $120 per room, and they rent an average of 40 rooms per night at this rate.

Next, we'll determine the number of rooms rented for each $10 reduction in price. According to the given information, for each $10 reduction in price, they rent an average of 10 more rooms.

So, for a $10 reduction in price, the number of rooms rented becomes 40 + 10 = 50. Similarly, for a $20 reduction in price, the number of rooms rented becomes 40 + 20 = 60, and so on.

Now, to find the rate they should charge to maximize revenue, we need to consider the relationship between price, number of rooms rented, and revenue.

Revenue can be calculated by multiplying the price per room by the number of rooms rented. Thus, the revenue can be expressed as:

Revenue = Price * Number of Rooms

Let's calculate the revenue for each scenario:

1. Original price: $120 per room, Number of rooms rented: 40
Revenue = $120 * 40 = $4800

2. $10 reduction in price: $120 - $10 = $110 per room, Number of rooms rented: 50
Revenue = $110 * 50 = $5500

3. $20 reduction in price: $120 - $20 = $100 per room, Number of rooms rented: 60
Revenue = $100 * 60 = $6000

From this analysis, we can see that the revenue increases as the number of rooms rented increases. However, reducing the price also reduces the revenue per room.

To find the rate that maximizes revenue, we need to find the point where the increase in the number of rooms rented compensates for the decrease in the price per room.

From the given information, we can assume that the trend of 10 more rooms rented for every $10 reduction in price continues.

To maximize revenue, we should continue reducing the price until the revenue no longer increases significantly.

Let's calculate the revenue for further reductions:

4. $30 reduction in price: $120 - $30 = $90 per room, Number of rooms rented: 70
Revenue = $90 * 70 = $6300

5. $40 reduction in price: $120 - $40 = $80 per room, Number of rooms rented: 80
Revenue = $80 * 80 = $6400

As the number of rooms rented continues to increase, the revenue per room starts to decrease significantly. Therefore, charging $80 per room would maximize revenue.

Hence, the rate they should be charging to maximize revenue is $80 per room.

To find the rate that the hotel chain should be charging to maximize revenue, we need to analyze the relationship between the price per room and the number of rooms rented. Let's break down the problem step by step.

1. Calculate the revenue at the current price:
Revenue = Price per room * Number of rooms rented
Revenue = $120 * 40
Revenue = $4,800

2. Calculate the revenue for each $10 reduction in price:
For each $10 reduction in price, 10 more rooms are rented. Therefore, the additional revenue for each $10 reduction in price is:
Additional Revenue = Price reduction * Additional rooms rented
Additional Revenue = $10 * 10
Additional Revenue = $100

3. Find the price reduction that maximizes revenue:
If the hotel reduces the price by $10, the revenue increases by $100. To maximize revenue, we want to find the price reduction that maximizes the additional revenue (i.e., $100). Beyond that, if the price is reduced further, the hotel may not generate enough additional demand to compensate for the reduced price. Therefore, we need to determine how many times the price needs to be reduced by $10 to maximize the additional revenue.

Let's assume the price per room after the reduction is P.
The additional revenue can be represented as a function of the price reduction (R):
Additional Revenue = R * (10 + 10 * R)
Additional Revenue = 10R + 10R^2

We differentiate this function with respect to R, set it equal to zero, and solve for R to find the maximum additional revenue.

d(Additional Revenue)/dR = 10 + 20R

Setting it equal to zero:
10 + 20R = 0
R = -0.5

Since the price reduction (R) cannot be negative, we ignore the negative value (-0.5). Therefore, the price reduction that maximizes the additional revenue is zero (no reduction in price).

4. Calculate the new price per room:
The price per room after the reduction is P = $120 + $10 * 0 = $120

5. Calculate the revenue at the new price:
Revenue = Price per room * Number of rooms rented
Revenue = $120 * (40 + 10 * 0)
Revenue = $120 * 40
Revenue = $4,800

Therefore, to maximize revenue, the hotel chain should be charging $120 per room.

Please note that this analysis assumes that the relationship between price and demand remains consistent. In reality, factors like competition, location, time of the year, and other market dynamics can impact the optimal price.