022 Exam 2 Sample A, Problem 7

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Find the quantity that produces maximum profit, given the demand function and cost function .

Foundations:  
Recall that the demand function, , relates the price per unit to the number of units sold, .

Moreover, we have several important important functions:

  • , the total cost to produce units;
  • , the total revenue (or gross receipts) from producing units;
  • , the total profit from producing units.
In particular, we have the relations
and
Using these equations, we can find the maximizing production level by determining when the first derivative of profit is zero.

 Solution:

Step 1:  
Find the Profit Function: We have
From this,
Step 2:  
Find the Maximum: The equation for marginal revenue is
Applying our power rule to each term, we find
The only root of this occurs at , and this is our production level to achieve maximum profit.
Final Answer:  
Maximum profit occurs when we produce 15 items.


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