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.)
$?
To find the maximum profit, we need to determine the number of bottles beyond the initial 9600 that will result in the highest total profit.
Let's start by calculating the total profit for the initial 9600 bottles:
Profit for the first 9600 bottles = 9600 * $5 = $48,000
To find the number of bottles beyond 9600 that will maximize the profit, we need to determine the point at which the profit per bottle drops below $0.0002.
Let's assume the number of bottles beyond 9600 is x.
The profit for these extra bottles can be represented by the following equation:
Profit beyond 9600 bottles = x * (5 - 0.0002x)
To find the maximum profit, we need to find the value of x that maximizes this equation. One way to do this is by finding the vertex of the quadratic function:
The vertex of the quadratic function ax^2 + bx + c can be determined using the formula:
x = -b / (2a)
In our case, a = -0.0002, and b = 5.
x = -5 / (2 * -0.0002)
x = -5 / -0.0004
x = 12500
Rounding the number of bottles down to the nearest whole bottle:
Number of bottles for maximum profit = 12500 (rounded down)
To calculate the maximum profit, we can substitute this value back into the profit equation:
Profit beyond 9600 bottles = 12500 * (5 - 0.0002 * 12500)
Profit beyond 9600 bottles = 12500 * (5 - 2.5)
Profit beyond 9600 bottles = 12500 * 2.5
Profit beyond 9600 bottles = $31,250
Adding this profit to the profit from the initial 9600 bottles:
Total maximum profit = $48,000 + $31,250
Total maximum profit = $79,250
Therefore, the maximum profit is $79,250.
To find the profit per bottle in this case, we divide the total maximum profit by the total number of bottles:
Profit per bottle = $79,250 / (9600 + 12500)
Profit per bottle ≈ $1.99
Therefore, the profit per bottle in this case is approximately $1.99.