A retired potter can produce china pitchers at a cost of 500

A retired potter can produce china pitchers at a cost of $5.00 each. Shes estimates her price function to be p=17-0.5x, where p is the price at which exactly x pitchers will be sold per week. Find the number of pitchers that shw should produce and the price that she should charge to maximize profit, and find her maximum profit.

Solution

the potter can produce pitchers at a cost of $5. The price function is p = 17 - 0.5*x where x is the number of pitchers sold in a week. The profit made by the potter when x pitchers are sold is P = px - 5x = (17 - 0.5x)x - 5x = 17x - 0.5x^2 - 5x = 12x - 0.5x^2

To maximize profit, the number of pitchers that should be sold is given by the solution of P\' = 0

=> 12 - x = 0

=> x = 12

The price of the pitchers at which 12 are sold is 17 - 6 = 11. The maximum profit made by the potter is $72

To maximize profits, the potter should charge $11 per pitcher and the maximum profit is $72