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At What Price? by Shermane King

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The Cookie Problem "The Fresh Bakery currently sells 1000 large chocolate chip cookies each week for $0.50 each. The bakery would like to increase its revenue from the cookie sales by increasing this price. A survey convinced the manager that for every $0.10 increase, the bakery would sell 70 fewer cookies each week. At what price should the cookies be sold to maximize the revenue to the bakery?”

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**Initial Solution Algebra I: Create a table**

Create an equation for cookies, price and revenue: 𝐶=1000−70𝑑 , 𝑃= 𝑑, and 𝑅=𝐶𝑃 (R = Cookies*Price) Utilize that equations for the rest of the data and analyze the table to determine the price that will maximize revenue. x cookies price revenue 1000 $ $ 1 930 $ $ 2 860 $ $ 3 790 $ $ 4 720 $ $ 5 650 $ $ 6 580 $ $ 7 510 $ $ 8 440 $ $ 9 370 $ $

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**Solution!! Pre-Calculus 𝑅=−7 4.64 2 +65 4.64 +500=650.89**

Know the equations for each: 𝑃= 𝑑, 𝐷=1000−70𝑑, and 𝑅=𝑃𝐷 The revenue equation is a quadratic function 𝑅(𝑑)=−7 𝑑 2 +65𝑑+500 Graph the equation, which is a parabola Find the vertex of the parabola which is d=65/14=4.64. 4.64 represent the increase in dimes, so the cookie price is $0.50+$0.10∗4.64=$0.96 𝑅=− =650.89

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**Connection to Secondary Mathematics**

Algebra I: create a table, analyze table, generate solution. Algebra II: find the vertex of the parabola. Pre-Calculus: locate the vertex of the quadratic 𝑦=𝑎𝑥²+𝑏𝑥+𝑐 which is ℎ= −𝑏 2𝑎 . Calculus: find first derivative and than solve for zero.

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**Extensions Given revenue find the price and amount of cookies Cost**

Profit = revenue – cost Change the problem completely

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