Phillip, the proprietor of a vineyard, estimates that the first 10400 bottles of wine produced this season will fetch a profit of $5 per bottle. However, the profit from each bottle beyond 10400 drops by $0.0003 for each additional bottle sold. Assuming at least 10400 bottles of wine are produced and sold, what is the maximum profit?

Question

Mathematics

Frosty

29 October, 02:16

Phillip, the proprietor of a vineyard, estimates that the first 10400 bottles of wine produced this season will fetch a profit of $5 per bottle. However, the profit from each bottle beyond 10400 drops by $0.0003 for each additional bottle sold. Assuming at least 10400 bottles of wine are produced and sold, what is the maximum profit?

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Kailee Weiss · Answer 1

The first 10900 bottles make a profit of 10900 * $5 = $54500

After that, the profit is:

$4.9997 + $4.9994 + $4.9991 + ...

And that pattern continues until it reaches zero (no profit earned).

Since this is an arithmetic series, the number of terms in the series is $5 / 0.0003, which is around 83333

Multiply that by the average term ($2.50), and you get a profit of around $208333.33

Add that to the original $54500 and you get $262833.33