Question 228146
 Suppose you have a lemonade stand, and when you charge $2 per cup of lemonade you sell 120 cups. But when you raise your price to $3 you only sell 60 cups. 
a. Write an equation for the number of cups you sell as a function of the price you charge.
You have two points relating cups sold and price: (120,2)(60,3)
slope = (2-3)/(120-60) = -1/60
Since 2 = (-1/60)*120+b, b = 2+2 = 4
Equation: 
P = (-1/60)c + 4
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b. Denote "C" for number of cups, and "P" for the price you charge. 
c. Assume the function is linear.
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3. Continuing our lemonade stand question: 
a. We all know that total revenue (TR) is a function of the price we charged (P) multiplied by the item quantity sold (in our case – Cups), i.e., TR = Price * Cups 
b. Please write the equation for your TR by inputting your answer from the function you have calculated in question #2.
TR = c*[(-1/60)c+4]
TR = (-1/60)c^2 + 4c
Note: This is a quadratic with a = (-1/60) and b = 4
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c. What price would maximize your TR? 
Max occurs when c = -b/2a = -4/(-2/60) = 120 cups
P(120) = $2 so a price of $2 gives you maximum revenue.
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Cheers,
Stan H.