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​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.

Respuesta :

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 - 4x

Let 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 - 4x²

To maximize the profit, [tex]\frac{dP}{dx}=0[/tex]

P' = 17 - 8x = 0

8x = 17

[tex]x=\frac{17}{8}[/tex]

x = 2.125

x ≈ 2

Therefore, 2 pitchers should be produced.

Profit = 17x - 4x²

         = 17(2) - 4(2)²

         = 34 - 16

         = $18

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