Maximum profit

Question

Phillip, the proprietor of a vineyard, estimates that the first 10000 bottles of wine produced this season will fetch a profit of $3 per bottle. However, the profit from each bottle beyond 10000 drops by $0.0002 for each additional bottle sold. Assuming at least 10000 bottles of wine are produced and sold, what is the maximum profit? (Round your answer correct to the nearest cent.)
$


What would be the profit/bottle in this case? (Round the number of bottles down to the nearest whole bottle. Round your answer correct to the nearest cent.)
$

Answer 1

Let x = the number of bottles produced above 10,000.  Hence the total number of bottles produced in 10,000 + x

 

Profit = (10,000 + x)($3 - $0.0002x)

Profit = 30,000 - 2x + 3x - 0.0002x²

 

Take the derivative of the Profit wrt x, set it to zero to find the extreme (maximum) point, and solve for x. The total number of bottles is 10,000 + x. Plug the value of x into the profit equation to find the profit. Divide by the number of bottles (10,000+x) to find the profit per bottle.