Let's start by calculating the store's revenue at the current price of $50 per pair of swimming flippers:
Revenue = Price x Quantity Sold = $50 x 92 = $4,600 per day
Now, let's see how changes in the price affect the quantity sold. According to the problem, for each $3 increase in price, 3 fewer sales are made. This means that the demand function is:
Quantity Sold = 92 - 3/3 (Price - $50) = 92 - (Price - $50)
where Price is measured in dollars.
To calculate the store's profit, we need to subtract the cost of producing each pair of swimming flippers from the revenue:
Profit = (Price - Cost) x Quantity Sold
We don't have information about the cost of producing each pair of swimming flippers, so let's assume that it is a constant of $20 per pair. This means that the profit function is:
Profit = (Price - $20) x (92 - (Price - $50)) = (Price - $20) x (-Price + $142)
Expanding the brackets and simplifying, we get:
Profit = -$Price^2 + $122Price - $2840
To find the price that maximizes profit, we need to take the derivative of the profit function with respect to price, and set it equal to zero:
dProfit/dPrice = -$2Price + $122 = 0
Solving for Price, we get:
Price = $61
So, the store should charge $61 per pair of swimming flippers to maximize profit. To verify that this is indeed the maximum, we can take the second derivative of the profit function with respect to price:
d^2Profit/dPrice^2 = -$2
Since this is negative, we know that the profit function is concave down, which means that the critical point we found is indeed a maximum.