Question

A hotel rents rooms to customers by the night. The hotel determines that if it sets the price of the room to be​ $160 per​ night,140

rooms will be rented. In order to rent 190 rooms, it must lower the price to ​$110 per night. What price must it set in order to maximize​ revenue?
The price would be ​$enter your response here per night.

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Answer 1

To maximize revenue, we need to find the price that will result in the highest number of rented rooms. We can do this by finding the point at which the slope of the revenue function is equal to zero.

Step-by-step explanation:

To maximize revenue, we need to find the price that will result in the highest number of rented rooms.

First, let's find the slope of the demand curve by calculating the change in the number of rented rooms and the change in price: (190-140)/(110-160) = 50/(-50) = -1.

Since the slope is negative, we know that as the price decreases, the number of rented rooms increases. To find the price that maximizes revenue, we need to find the point at which the slope of the revenue function is equal to zero. In this case, the revenue function is price multiplied by quantity.

The revenue function is given by R = price * quantity. Considering the demand curve, we can express quantity in terms of price: price = -1 * quantity + 200. Now we can substitute this expression into the revenue function: R = ( -1 * quantity + 200 ) * quantity.

To find the maximum revenue, we can take the derivative of the revenue function with respect to quantity and set it equal to zero: dR/dquantity = -2q + 200 = 0. Solving this equation, we find that the quantity that maximizes revenue is q = 100. Substituting this back into the demand curve expression, we can find the corresponding price: price = -1 * 100 + 200 = $100.

Therefore, the price that maximizes revenue is $100 per night.