   Chapter 2.3, Problem 83E

Chapter
Section
Textbook Problem

Profit of a Vineyard Phillip, the proprietor of a vineyard, estimates that if 10,000 bottles of wine were produced this season, then the profit would be $5/bottle. But if more than 10,000 bottles were produced, then the profit per bottle for the entire lot would drop by$0.0002 for each additional bottle sold.Assume that at least 10,000 bottles of wine are produced and sold, and let x denote the number of bottles produced and sold in excess of 10,000. a. Find a function P giving the profit in terms of x. b. What is the profit Phillip can expect from the sale of 16,000 bottles of wine from his vineyard?

(a)

To determine
The function P that gives the profit in the variable of x.

Explanation

Given information:

The profit would be $5/bottle if 10,000 bottles are produced. If more than 10,000 bottles are produced, then the profit per bottle would drop by$0.0002 for each additional bottle sold.

Calculation:

Let x be the number of bottles produced and sold in excess of 10,000, then the total number of bottles produced and sold becomes 10,000+x .

The profit per bottle is $5, for each bottle sold in excess of 10,000, the profit decreases by$0.0002 per bottle, so the profit per bottle becomes (50.0002x)

(b)

To determine
The profit for the sale of 16,000 bottles.

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