Phillip, the proprietor of a vineyard, estimates that the first 10900 bottles of

Question

Phillip, the proprietor of a vineyard, estimates that the first 10900 bottles of wine produced this season will fetch a profit of $4 per bottle. However, the profit from each bottle beyond 10900 drops by $0.0003 for each additional bottle sold. Assuming at least 10900 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.)
$ ?

Explanation / Answer

The profit in this case is given by the number of bottles of wine sold multiplied by the profit per bottle sold.

The profit per bottle is given by f(x) = 4 0.0003(x 10900),

so the total profit is given by : p(x) = xf(x) = [7.27x - 0.0003x2]

We now need to maximize this function with respect to x.

Following our system : p'(x) = 7.27 - 0.0006x

put p'(x) = 0 : thus x = 7.27/0.0006 => 12116.6667

Substituting the endpoint of 10900 and the critical point of 12116.667

p(10900) => 43600

p(12116.667) => 44044.0833

Thus they should sell 12116.667 bottles, at a profit per bottle of

f(12116.667) : $3.6349999

and total profit of $44044.0833

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