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.)
$ ?
profit/bottle is 5.00 - 0.0002(x-9600) for x bottles, x > 9600
total profit is thus
p(x) = 5*9600 + (x-9600)(5.00 - 0.0002(x-9600))
= -.0002x^2 + 8.84x - 18432
This is a parabola opening downward, with vertex at x = 22100
p(22100) = 79250
To find the maximum profit, we need to determine the number of bottles that will maximize the profit and calculate the corresponding profit per bottle.
Let's start by finding the number of bottles beyond which the profit per bottle decreases by $0.0002 for each additional bottle sold.
Let x be the number of bottles beyond 9600.
We can set up the equation:
Profit = (9600 x $5) + (x^2 x -$0.0002)
To find the maximum profit, we need to maximize this equation. To do this, we can take the derivative of the equation with respect to x and set it equal to zero:
Profit' = 0
Taking the derivative:
0 = 9600(0) + 2x(-$0.0002)
0 = -0.0004x
Setting x equal to zero:
0 = -0.0004x
x = 0
Since x represents the number of additional bottles sold, it cannot be zero. Therefore, there is no maximum profit beyond 9600 bottles.
To find the maximum profit, we calculate the profit from the first 9600 bottles:
Profit = 9600 x $5 = $48,000
Therefore, the maximum profit is $48,000.
Now, let's determine the profit per bottle.
To find the profit per bottle, we divide the maximum profit by the total number of bottles produced:
Profit per bottle = Maximum Profit / Total number of bottles
Profit per bottle = $48,000 / 9600
Profit per bottle ≈ $5.00
Therefore, the profit per bottle in this case is approximately $5.00.
To find the maximum profit, we need to determine the number of bottles at which the profit per bottle is the highest.
Let's analyze the situation step by step.
Step 1: Calculate the profit for the first 9600 bottles.
Profit for the first 9600 bottles = 9600 bottles * $5/bottle = $48,000
Step 2: Determine the number of bottles beyond 9600 at which the profit starts to decrease.
To find the number of bottles at which the profit starts to decrease, we need to determine when the profit per bottle drops below $5/bottle.
Let's calculate the number of bottles at which the profit per bottle drops below $5/bottle.
Profit per bottle beyond 9600 bottles = $5 - $0.0002 * (number of bottles beyond 9600)
Setting the profit per bottle to less than $5/bottle:
$5 - $0.0002 * (number of bottles beyond 9600) < $5
Simplifying the inequality:
- $0.0002 * (number of bottles beyond 9600) < 0
Dividing both sides by -0.0002 flips the inequality:
(number of bottles beyond 9600) > 0 / -0.0002
Solving:
(number of bottles beyond 9600) > 0
Since the number of bottles sold must be positive, we can conclude that beyond the first 9600 bottles, the profit per bottle will be less than $5/bottle.
Step 3: Calculate the profit beyond the first 9600 bottles.
As mentioned above, the profit per bottle beyond the first 9600 bottles decreases by $0.0002 for each additional bottle sold.
To find the maximum profit, we need to determine the number of bottles that can be sold beyond the first 9600, while still maintaining a positive profit. Since we know the profit per bottle decreases beyond this point, we need to calculate the number of bottles that can be sold before the profit per bottle reaches $0.
Let's calculate the maximum number of bottles that can be sold to maintain a positive profit:
Profit per bottle beyond 9600 bottles = $5 - $0.0002 * (number of bottles beyond 9600)
$5 - $0.0002 * (number of bottles beyond 9600) > 0
Simplifying the inequality:
$0.0002 * (number of bottles beyond 9600) < $5
(number of bottles beyond 9600) < $5 / $0.0002
Calculating the number of bottles:
(number of bottles beyond 9600) < 25000
Therefore, the maximum number of bottles that can be sold while maintaining a positive profit is 25,000.
Step 4: Calculate the maximum profit.
To find the maximum profit, we need to calculate the profit for the first 9600 bottles and the profit for the additional bottles.
Profit for the additional bottles = (number of bottles beyond 9600) * (profit per bottle beyond 9600)
Profit for the additional bottles = (25,000 - 9600) * ($5 - $0.0002 * (25,000 - 9600))
Therefore, the maximum profit would be:
Maximum profit = Profit for the first 9600 bottles + Profit for the additional bottles
Now that we have the equations set up, we can calculate the maximum profit and the profit per bottle in this case.