Phillip, the proprietor of a vineyard, estimates that the first 11000 bottles of
ID: 2837765 • Letter: P
Question
Phillip, the proprietor of a vineyard, estimates that the first 11000 bottles of wine produced this season will fetch a profit of $3 per bottle. However, the profit from each bottle beyond 11000 drops by $0.0001 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? (Round the number of bottles down to the nearest whole bottle. Round your answer correct to the nearest cent.) $Explanation / Answer
Hi
Since 11000 btls sold at $3 that profit is 33000
Profit function after 11000 btls
P(x)=33000+(3-.0001x)x
= 33000+3x-.0001x^2
DIfferentiating wrt x
P'(x) = 3 - .0002x
Since the max is reached when P'(x) = 0
3 - .0002x = 0
x = 15000
So the max profit is obtained when they sell 15000 bottles.
The max profit is P(15000) = 55500 dollars
Part 2
The profit per bottle at this point is given by 3 - .0001x. Lets call it f(x)
So f(15000) = 1.5
So $1.50 per bottle.