To determine the optimal price for each candle that maximizes the retailer's profit, we need to consider the relationship between the price and the number of candles sold.
Let's start by calculating the total revenue generated from selling the candles at the original price of $30 per candle. Since the retailer sold 120 candles, the total revenue is:
Total Revenue = Price per Candle × Number of Candles Sold
Total Revenue = $30 × 120
Total Revenue = $3600
Next, we need to consider the relationship between the price decrease and the increase in the number of candles sold. According to the information provided, for each $2 decrease in price, the retailer will sell 10 additional candles. This implies that the number of additional candles sold per $2 decrease in price has a constant ratio of 10/2 = 5.
To find the optimal price, we need to identify the point at which the increase in revenue from selling additional candles due to the price decrease is greater than the decrease in revenue resulting from the lower price. In other words, we need to find the price that maximizes the retailer's profit.
Let's assume the retailer decreases the price by $2 x times. The new price per candle will be $30 - ($2 × x). The additional number of candles sold will be 10 × x.
The new revenue generated from selling the additional candles is:
Additional Revenue = (Price per Candle - Price Decrease) × Additional Candles Sold
Additional Revenue = ($30 - ($2 × x)) × (10 × x)
Additional Revenue = (30 - 2x) × 10x
Additional Revenue = 300x - 20x^2
The retailer's total revenue after the price decrease will be the sum of the original revenue and the additional revenue:
Total Revenue = Original Revenue + Additional Revenue
Total Revenue = $3600 + (300x - 20x^2)
To maximize profit, we need to determine the price decrease that results in the highest total revenue. This can be achieved by finding the value of x that maximizes the Total Revenue function.
To find the optimal value of x, we can take the derivative of the Total Revenue function with respect to x and set it equal to zero:
d(Total Revenue) / dx = 0
d(3600 + 300x - 20x^2) / dx = 0
300 - 40x = 0
Solving the equation:
40x = 300
x = 300 / 40
x = 7.5
Since x represents the number of $2 price decreases, we multiply it by $2 to determine the total price decrease:
Total Price Decrease = $2 × x
Total Price Decrease = $2 × 7.5
Total Price Decrease = $15
The optimal price for each candle that maximizes the retailer's profit is the original price minus the total price decrease:
Optimal Price = Original Price - Total Price Decrease
Optimal Price = $30 - $15
Optimal Price = $15
Therefore, the retailer should charge $15 per candle in order to maximize its profit.