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.)
$ ?
Here's a start - not sure about the formula. Repost so someone answeres. $0.0002 x = $5 --> You can produce the original 9,600 bottles and an additional 25,000 before your profit deminishes to zero. Not sure what the formula would be to calculate the total value - you can't numerate for 9600 bottles...that's insane. So you would get an initial (9,600 x $5) then add on for each additional bottle produced +(1 x $4.9998) + (1 x $4.9996)....not sure how you would do this. Then, assume there are 12 bottles/case so (9600+25,000) / 12 = no. of cases. Then get your cost from the 1st part and divide by the number of cases.
To find the maximum profit, we need to determine the number of bottles beyond 9600 that should be sold to maximize profit.
First, let's find the maximum number of bottles of wine that can be sold. To do this, we need to determine when the profit per bottle drops to $0.
Given:
Profit per bottle beyond 9600 drops by $0.0002 for each additional bottle sold.
Let's set up an equation to find the maximum number of bottles, x, that can be sold:
Profit per bottle = $5 - $0.0002(x - 9600)
Setting the profit per bottle to $0:
0 = $5 - $0.0002(x - 9600)
To solve for x, we can rearrange the equation:
$0.0002(x - 9600) = $5
Divide both sides of the equation by $0.0002:
x - 9600 = $5 / $0.0002
x = ($5 / $0.0002) + 9600
x ≈ 24500
So, the maximum number of bottles that can be sold is approximately 24500 bottles.
Now, let's calculate the maximum profit. We know that the profit for the first 9600 bottles is $5 per bottle.
Profit for the first 9600 bottles = $5/bottle x 9600 bottles = $48000
For the remaining bottles beyond 9600, the profit decreases by $0.0002 for each additional bottle sold. So, the profit per bottle for the remaining bottles can be calculated using the formula:
Profit per bottle = $5 - $0.0002(x - 9600)
Since the maximum number of bottles that can be sold is approximately 24500, we can calculate the profit for the remaining bottles:
Profit for the remaining bottles = ($5 - $0.0002(x - 9600)) x (x - 9600)
Profit for the remaining bottles = ($5 - $0.0002(24500 - 9600)) x (24500 - 9600)
Profit for the remaining bottles ≈ ($5 - $0.0002(14900)) x (14900)
Profit for the remaining bottles ≈ ($5 - $2.98) x (14900)
Profit for the remaining bottles ≈ $2.02 x 14900
Profit for the remaining bottles ≈ $30298
Now, we can calculate the maximum profit by adding the profit for the first 9600 bottles to the profit for the remaining bottles:
Maximum profit ≈ $48000 + $30298 ≈ $78298
Therefore, the maximum profit is approximately $78298.
To find the profit per bottle, we can divide the maximum profit by the total number of bottles sold, which is 24500.
Profit per bottle ≈ $78298 / 24500 ≈ $3.19
Therefore, the profit per bottle in this case is approximately $3.19.