An artisan makes and sells necklaces on the beach during the summer. The material for each necklace costs him $200 and he has been selling approximately 20 necklaces a day for $600 each. Now you've been wondering whether or not you should raise the price, so you do a survey and discover that for every $50 increase in price you'll lose two sales a day. What price should you set for the necklaces in order to maximize daily profits?

To determine the price that maximizes daily profits, we need to analyze the relationship between price, sales, and costs. Let's start by calculating the current daily profit. The material cost for each necklace is $200, and the artisan sells approximately 20 necklaces a day for $600 each. Current daily revenue = Number of necklaces sold * Price per necklace Current daily revenue = 20 * $600 = $12,000 Current daily cost = Number of necklaces sold * Material cost per necklace Current daily cost = 20 * $200 = $4,000 Current daily profit = Current daily revenue - Current daily cost Current daily profit = $12,000 - $4,000 = $8,000 Now, let's consider the potential impact of increasing the price by $50. For every $50 increase, the artisan will lose two sales a day. Let's denote: P = Current price per necklace S = Number of necklaces sold per day D = Price increase per necklace L = Number of sales lost per day per price increase For our case: P = $600 S = 20 D = $50 L = 2 To find the optimal price that maximizes daily profits, we need to compare the profits for different price levels. We can calculate the profit for each price level and choose the highest value. Profit at current price: $8,000 Profit at increased price (P + D): Revenue = (S - L) * (P + D) Cost = S * Material cost per necklace Profit = Revenue - Cost Profit at increased price = (S - L) * (P + D) - S * Material cost per necklace Let's calculate the profit at the increased price for each $50 increment and determine the maximum profit: Price: $600 Profit = (20 - 0) * ($600) - 20 * $200 = $8,000 Price: $650 Profit = (20 - 2) * ($650) - 18 * $200 = $8,200 Price: $700 Profit = (20 - 4) * ($700) - 16 * $200 = $8,200 Price: $750 Profit = (20 - 6) * ($750) - 14 * $200 = $8,200 Price: $800 Profit = (20 - 8) * ($800) - 12 * $200 = $8,000 As we can see, the maximum profit is achieved at a price of $650 or $700, both yielding a profit of $8,200. Therefore, to maximize daily profits, the artisan should set the price for the necklaces at either $650 or $700.