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A boy is selling lemonade throughout the hot summer. Suppose the number of cups sold is...

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user3666642 | eNoter

Posted May 20, 2013 at 3:12 PM via web

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A boy is selling lemonade throughout the hot summer. Suppose the number of cups sold is given by the function, `n(X)=x2^(-x)+2` , where the price, x, in dollars determines the number of cups sold per day, n, in hundreds.

a)  What is the price that maximizes the number of cups sold?

b)  How many cups are sold at this price?

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crmhaske | College Teacher | (Level 3) Associate Educator

Posted May 24, 2013 at 3:22 PM (Answer #2)

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In order to determine the maximum number of cups sold, we need to first find for what value of x (price) n(x) is maximized.  To do so, first take the derivative of n(x):

`n'(x)=2^(-x)-x2^(-x)ln(2)`

Now set the derivative to 0 and solve for x:

`0=2^(-x)-x2^(-x)ln(2)`

`0=1-xln(2)`

`x=1/ln(2)=1.44`

a) Therefore, if they are sold at $1.44 the number sold is maximized.

b) `n(1.44)=1.44*2^(-1.44)+2=2.53~~3`

Approximately 3 cups are sold at a price of $1.44

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