Assuming at least 11000 bottles of wine are produced and sold, what is the maximum profit? (Round your answer correct to the nearest cent.)

Question

 the first 11000 bottles of wine produced this season will fetch a profit of $4 per bottle. the profit from each bottle beyond 11000 drops by $0.0003 for each additional bottle sold. Assuming at least 11000 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?

Answer 1

Write the profit function. Let P(x) be the profit where "x" is the number of bottles produced/sold:

For the first 11000 bottles the profit is: 11000(4)

Beyond the first 11,000 the number of bottles sold is (x - 11000) and the profit per bottle is [4 - 0.0003(x - 11000)].

So P(x) = 11000(4) + (x - 11000)[4 - 0.0003(x - 11000)].

P(x) = 44000 + (x - 11000)(4 - 0.0003x + 3.3)

P(x) = 44000 + (x - 11000)(7.3 - 0.0003x)

P(x) = 44000 + 7.3x - 0.0003x² - 80300 + 3.3x

P(x) = - 0.0003x² + 10.6x - 36300

To maximize, take the derivative and set it equal to zero. This is a negative quadratic so it will have a max value (which is what we are being asked for).

P'(x) = -0.0006x + 10.6

-0.0006x + 10.6 = 0

0.0006x = 10.6

x = 17667 bottles

The profit when 17667 bottles are sold is P(17667) = - 0.0003(17667)² + 10.6(17667) - 36300 = $57,333.33