Question:

Neighborhood entrepreneurs have found that the total cost function for their lemonade stand is \(C(x)=.0000015 x^{3}-.00045 x^{2}+.17 x+7.43\), including setting up the stand, making the lemonade, and serving it. The kids determine that the demand function for their lemonade is \(p=1.2-.01 x\), where \(p\) is the price at which \(x\) servings of lemonade are demanded. Determine the price that the kids should charge to maximize their profit. How many servings of lemonade will be sold at that price? (You may round the number of servings to the nearest whole number.) What is the maximum profit? Use calculus techniques to solve this problem. State the domain of your function. Remember to verify that you've found a maximum by using either the First Derivative Test or the Second Derivative Test, or by checking for absolute extrema.

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