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Each of the functions is a company's price function, where p is the price (in dollars)...

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jjmgingrich | Student, Undergraduate | (Level 1) Salutatorian

Posted July 3, 2013 at 2:43 AM via web

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Each of the functions is a company's price function, where p is the price (in dollars) at which quantity x (in thousands) will be sold. 1. Find the revenue function R(x). 2. Find the quantity and price that will maximize revenue. When p=4-ln x

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mjripalda | High School Teacher | (Level 1) Senior Educator

Posted July 3, 2013 at 3:27 AM (Answer #1)

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(1) To determine the revenue function, apply the formula:

Revenue= Price per unit * Number of units sold

Since the price per unit is p and the number of units sold is x, then:


Plug-in p=4-lnx.


Hence, the revenue function is` R(x)=(4-lnx)x ` .

(2) To solve for the values of x and p that will maximize the revenue,take the derivative of R(x). Apply the product rule which is `(u*v)=u*v'+v*u` .

`R'(x)=(4-lnx)*x' + x*(4-lnx)'`




Then, set R'(x) equal to zero and solve for x.




And convert this to exponential equation to get x only at one side of the equation.
Note that the equivalent exponential form of `ln m = a`  is ` m=e^a` .



Since x refers to the number of units sold, round it off to the next higher whole number.


Now that the value of x is known, plug-in this to p=4-lnx .



Hence, 21 units must be sold at $1 each to maximize the revenue.

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