Mila sells CDs at $20 each. At this price she is able to sell 280 CDs per day. However, market research shows that for every $0.50 increase in price, her sales drop by 5 units. At what sale price should she set her CDs to be in order to maximize revenue? What is her maximum revenue at this price?
Solve this in detailed
To find the sale price that will maximize revenue, we need to consider the relationship between price, quantity sold, and revenue.
Let's start by assuming that Mila sets her CDs at the initial price of $20. At this price, she is able to sell 280 CDs per day. Therefore, her revenue at this price is 280 CDs multiplied by $20:
Revenue at $20 = 280 CDs * $20 = $5600
Now let's consider the relationship between price and quantity sold. We know that for every $0.50 increase in price, her sales drop by 5 units. This means that for each unit increase in price, the quantity sold decreases by 5/0.50 = 10 units.
To determine the optimal price, we need to find the price at which the quantity sold multiplied by the price will give us the maximum revenue.
Let's assume that Mila increases the price by 0.50 and the quantity sold decreases by 10 units. We can calculate the revenue at the new price using the following formula:
Revenue = Quantity sold * Price
Revenue at the new price = (280 - 10) CDs * ($20 + 0.50) = 270 CDs * $20.50 = $5535
We can repeat this process by increasing the price and decreasing the quantity sold by 0.50 and 10 units respectively, and calculate the revenue each time:
Revenue at price increase of $0.50 = (270 - 10) CDs * ($20 + 0.50) = 260 CDs * $20.50 = $5330
Revenue at price increase of $1 = (260 - 10) CDs * ($20 + 0.50 * 2) = 250 CDs * $21 = $5250
Revenue at price increase of $1.50 = (250 - 10) CDs * ($20 + 0.50 * 3) = 240 CDs * $21.50 = $5160
We can continue this process until we reach a point where the revenue starts to decrease. However, it becomes apparent from the calculations above that the revenue is decreasing as we increase the price. Therefore, the optimal price to maximize revenue is $20.
The maximum revenue at this price is $5600.