Phillip, the proprietor of a vineyard, estimates that the first 9600 bottles of wine produced this season will fetch a profit of $5 per bottle. However, the profit from each bottle beyond 9600 drops by $0.0002 for each additional bottle sold. Assuming at least 9600 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.)
$
I need answer
To find the maximum profit, we need to determine how many bottles should be produced and sold beyond the initial 9600 bottles.
Let's calculate the profit for each additional bottle beyond 9600 using the given information. The profit from each bottle drops by $0.0002 for each additional bottle sold.
So, the profit for each additional bottle can be calculated as:
Profit/bottle = $5 - ($0.0002 × number of additional bottles)
We want to find the maximum profit, which means we need to find the point where the profit per bottle drops to zero.
Setting the Profit/bottle equation to zero:
$5 - ($0.0002 × number of additional bottles) = 0
Now, let's solve for the number of additional bottles:
$0.0002 × number of additional bottles = $5
number of additional bottles = $5 / $0.0002
number of additional bottles = 25000
Since we want to find the maximum profit, we add the number of additional bottles to the initial 9600 bottles:
Total number of bottles = 9600 + 25000
Total number of bottles = 34600
Now, let's calculate the profit per bottle for this case:
Profit per bottle = $5 - ($0.0002 × 25000)
Profit per bottle = $5 - $5
Profit per bottle = $0
So, beyond 34600 bottles, the profit per bottle becomes zero.
Now, let's calculate the maximum profit by multiplying the profit per bottle by the total number of bottles:
Maximum profit = Profit per bottle × Total number of bottles
Maximum profit = $0 × 34600
Maximum profit = $0
Hence, the maximum profit in this case would be $0.