Optimization Calculus Question

Question

Phillip, the proprietor of a vineyard, estimates that the first 11000 bottles of wine produced this season will fetch a profit of $7 per bottle. However, the profit from each bottle beyond 11000 drops by $0.0001 for each additional bottle sold. Assuming at least 11000 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.) 

$ 

Answer 1

Hi Brendon!

The first thing we need to do to solve this problem is write a function for the profit based on the number of bottles. Then, we need to find the maximum of that function by taking the derivative and finding the zeros of the derivative function. After that, we plug the value at which the maximum occurs back into our profit function to calculate the maximum profit. Finally, we will calculate the profit per bottle by dividing the maximum profit by the number of bottles.

Let's start with the first step:

Write an equation for the profit based on the number of bottles

We know that the first 11,000 bottles will return $7 per bottle. This is guaranteed, as the profit doesn't diminish until we exceed 11,000 bottles. Thus, we know that we have a constant in the equation equivalent to (11,000)*($7) = $77,000

We also know that each bottle in excess of 11,000 will cause the profit to diminish by $0.0001

In other terms:

Profit after 11,000 bottles = x*($7 - $0.0001*x), where x is the number of bottles in excess of 11,000

Now we can combine the two parts into a total profit equation:

P = 77,000 + x*(7 - 0.0001*x)

Which can be written as:

P = -0.0001*x² + 7*x + 77,000

Now we are ready for the next step:

Find the maximum of that function by taking the derivative and finding the zeros of the derivative function

First, we take the derivative of the profit function with respect to the number of bottles in excess of 11,000, x. This can be done by using the power rule:

dP/dX = -0.0002*x + 7

Now, we set dP/dX equal to zero and solve the equation for the number of bottles in excess of 11,000:

0 = -0.0002*x + 7

0.0002*x = 7

35000 = x

After solving for x, we conclude that the maximum profit occurs when we sell 35,000 bottles more than 11,000 bottles, or rather, the maximum profit occurs when we sell 11,000+35,000=46,000 bottles. Even though the maximum occurs at x = 35000, our equation already had 11,000 bottles baked in, which is where the constant $77,000 came from.

Anyway, we're ready for the third step:

Plug the value at which the maximum occurs back into our profit function to calculate the maximum profit

This step is simple, we just need to plug the 35,000 back into our profit function. Even though we determined that the maximum profit occurs at 46,000 bottles, we will ignore 11,000 of them, as our profit function already includes the profit from these bottles.

Our profit function based on bottles in excess of 11,000 is:

P = -0.0001*x² + 7*x + 77,000

Plugging in 35,000...

P = -0.0001*(35000)² + 7*35000 + 77000

P = $199,500

Thus, the maximum profit is $199,500

Now, we can move on to the last step:

Calculate the profit per bottle

We determine the profit per bottle by dividing the maximum profit by the number of bottles.

We know that the maximum profit is $199500, and we know that the number of bottles to achieve this is 35,000+11,000 = 46,000 bottles

Thus, we can calculate the profit per bottle immediately:

Profit/bottle = ($199,500) / (46,000 bottles) = $4.3369

Rounding to the nearest cent:

Profit/bottle = $4.34

**Note: I assumed in this problem that each consecutive bottle would reduce the profit of all bottles in excess of 11,000 evenly, such that at 12,000 bottles, the price reduction would be 0.0001*1000=$0.1

If instead the price reduction is for each bottle separately (bottle 11,001 is $6.9999, bottle 11,002 is $6.9998, etc), then you will need to solve this problem with a different method. If the latter price function is applicable, let me know and I can walk you through it.

Answer 2

Phillip, the proprietor of a vineyard, estimates that the first 11000 bottles of wine produced this season will fetch a profit of $7 per bottle. However, the profit from each bottle beyond 11000 drops by $0.0001 for each additional bottle sold. Assuming at least 11000 bottles of wine are produced and sold, what is the maximum profit?

Every bottle sold will bring in more money so the profits keeps going up UNTIL the price of a bottle is $0.00.

Now $7 is 7000 * $.0001. So if you sold 6999 over the 1st 11000 you profit will still go up

For the 1st 11,000 you get $7. So that brings in $77,000.

The next 6999 will bring in 6.999 + 6.998 + 6.997 + ... + .01 = (1/1000)(6999 + 6998 + ... + 1)

Continue from here.