Amy, the owner of Amy's Pottery, can produce china pitchers at a cost of $ 3 each. She estimates the price function to be p equals 20 minus 4 x comma where p is the price at which exactly x pitchers will be sold per week. Find the number of pitchers that she should produce and the price that she should charge in order to maximize profit. Also, find the maximum profit.
Accepted Solution
A:
Answer:Amy should produce 2 pitchers to maximize profit and the maximum profit will be $18.Step-by-step explanation:Amy can produce china pitchers at a cost of $3 each.She estimates the price function that can be represented by p = 20 - 4xLet she sells the number of x pitchers in a week.Then the selling price of x pitchers will be = $x(20 - 4x)Cost price of x pitchers = $3x Profit (P) = Selling price - Cost price = x(20 - 4x) - 3x = 20x - 4x²- 3x = 17x - 4x²To maximize the profit, [tex]\frac{dP}{dx}=0[/tex]P' = 17 - 8x = 08x = 17[tex]x=\frac{17}{8}[/tex]x = 2.125x ≈ 2Therefore, 2 pitchers should be produced.Profit = 17x - 4x² = 17(2) - 4(2)² = 34 - 16 = $18