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.)
$ ?
well it'll be about 25 bottles sold before the profit changes; 5.00 will become approximately 4.99 after 25 bottles sold (0.005 rounds off to 0.01). after the first 25, the profit won't change to 4.98 until 50 bottles are sold. (0.015 will round off to 0.02) the 50 bottles per cycle will continue through out the process;
25 bottles sold = $125 profit ($5*25)
75 bottles sold = $125 + $249.5 ($4.99*50)
125 bottles sold = $374.5 + ($4.98*50)
continue this process and you will find the answer once you reach 9600 bottles.
For the second part; take the answer from the first question and divide it by 9600.
To find the maximum profit, we need to determine the number of bottles at which the profit starts decreasing.
Let's set up an equation to represent the profit per bottle beyond 9600 bottles:
Profit = $5 - ($0.0002 x Number of Bottles Beyond 9600)
To find the number of bottles at which the profit starts decreasing, we need to find where the profit per bottle becomes negative:
$5 - ($0.0002 x Number of Bottles Beyond 9600) < 0
$5 - $0.0002x < 0
- $0.0002x < - $5
Divide both sides by - $0.0002 (since it's a negative number, the inequality sign will change):
x > $5 / $0.0002
x > 25000
Therefore, the maximum profit will be achieved when the number of bottles sold is 25000 or more.
To calculate the maximum profit, we need to determine the profit from the first 9600 bottles, which is $5 per bottle:
Profit from the first 9600 bottles = $5 x 9600 = $48,000
Now, let's determine the profit from the bottles beyond 9600. Since the profit per bottle decreases linearly by $0.0002, we can use the formula for the sum of an arithmetic series to calculate the total profit:
Profit from the bottles beyond 9600 = (First Term + Last Term) x Number of Terms / 2
First Term = $5 - ($0.0002 x 9601) = $3.8398 (rounded to nearest cent)
Last Term = $5 - ($0.0002 x 25000) = $0
Number of Terms = 25000 - 9600 = 15400
Profit from the bottles beyond 9600 = ($3.8398 + $0) x 15400 / 2 = $29,707.38 (rounded to nearest cent)
Therefore, the maximum profit is the sum of the profit from the first 9600 bottles and the profit from the bottles beyond 9600:
Maximum Profit = $48,000 + $29,707.38 = $77,707.38
So, the maximum profit is $77,707.38.
To find the profit per bottle in this case, we need to divide the maximum profit by the total number of bottles sold. Since we know that at least 9600 bottles are sold, we can use the maximum profit calculation for that scenario.
Profit per bottle = Maximum Profit / Number of Bottles
Number of Bottles = 9600
Profit per bottle = $77,707.38 / 9600 = $8.08
Therefore, the profit per bottle in this case is $8.08.
To calculate the maximum profit, we need to determine the number of bottles at which the profit per bottle drops to zero. This occurs when the profit from each additional bottle sold equals the decrease in profit of $0.0002.
Let's break down the problem step by step:
1. Start with the initial profit per bottle:
The first 9600 bottles fetch a profit of $5 per bottle.
2. Determine the maximum number of bottles where the profit decreases:
Let's assume this maximum number of bottles is x.
So, the profit from the (9601)st bottle is $4.9998 ($5 - $0.0002).
The profit from the (9602)nd bottle is $4.9996 ($5 - 2*$0.0002).
The profit from the (9603)rd bottle is $4.9994 ($5 - 3*$0.0002).
We can see the pattern: the profit from the (9600+n)th bottle is $5 - n*$0.0002.
3. Calculate the profit function for any given number of bottles (b):
- If 9600 < b <= x:
The profit per bottle is $5 - (b - 9600) * $0.0002.
- If b > x:
The profit per bottle is $0.
4. Determine the maximum profit:
To find the maximum profit, we need to calculate the total profit for each case:
- If 9600 < b <= x:
The total profit is the sum of the profit per bottle multiplied by the number of bottles: Total Profit = (b - 9600) * ($5 - (b - 9600) * $0.0002).
- If b > x:
The total profit is 0 since the profit per bottle drops to $0.
5. Find the value of x:
We need to find the number of bottles at which the profit per bottle drops to zero.
So, we set $5 - (b - 9600) * $0.0002 = 0 and solve for b:
$5 - (b - 9600) * $0.0002 = 0.
Simplifying and solving, we get b = 10990.
6. Calculate the maximum profit:
We know that the maximum profit occurs when b > x, so the maximum profit is 0.
Therefore, the maximum profit is $0.
To determine the profit per bottle in this case, we can use the formula derived above: $5 - (b - 9600) * $0.0002.
Substituting b = 9600 (rounded down to the nearest whole bottle), we can calculate the profit per bottle:
Profit per bottle = $5 - (9600 - 9600) * $0.0002 = $5.
Therefore, the profit per bottle in this case is $5.