Let's work through the given problem step-by-step.
### Step 1: Find the Profit Function
Given:
- Cost function, [tex]\( C(x) = 0.2x + 31 \)[/tex]
- Price-demand function, [tex]\( p(x) = 6.756 - 0.002x \)[/tex]
First, we need to find the revenue function, [tex]\( R(x) \)[/tex], which is the product of the number of containers, [tex]\( x \)[/tex], and the price per container, [tex]\( p(x) \)[/tex]:
[tex]\[ R(x) = p(x) \cdot x = (6.756 - 0.002x) \cdot x \][/tex]
[tex]\[ R(x) = 6.756x - 0.002x^2 \][/tex]
Next, we find the profit function, [tex]\( P(x) \)[/tex], which is the revenue minus the cost:
[tex]\[ P(x) = R(x) - C(x) = (6.756x - 0.002x^2) - (0.2x + 31) \][/tex]
[tex]\[ P(x) = 6.756x - 0.002x^2 - 0.2x - 31 \][/tex]
[tex]\[ P(x) = -0.002x^2 + 6.556x - 31 \][/tex]
Thus, the profit function is:
[tex]\[ P(x) = -0.002x^2 + 6.556x - 31 \][/tex]
### Step 2: Number of Containers to Maximize Profit
To find the number of containers that maximize the profit, we need to find the critical points by setting the first derivative of the profit function to zero and solving for [tex]\( x \)[/tex]. We then need to determine which of these points is a maximum by applying the second derivative test.
The result from our calculations is that the number of containers that maximizes profit is:
[tex]\[ x = 1639 \][/tex]
### Step 3: Find the Maximum Profit
We now substitute the value [tex]\( x = 1639 \)[/tex] back into the profit function to find the maximum profit.
The maximum profit is:
[tex]\[ P(1639) = 5341.642 \, \text{dollars} \][/tex]
### Step 4: Find the Price to Charge Per Container
Finally, we need to find the price per container at the point where the profit is maximized. We substitute [tex]\( x = 1639 \)[/tex] into the price-demand function [tex]\( p(x) \)[/tex]:
[tex]\[ p(1639) = 6.756 - 0.002 \cdot 1639 \][/tex]
[tex]\[ p(1639) = 3.478 \, \text{dollars} \][/tex]
### Summary Answers
Let's summarize the answers to the questions:
Profit Function:
[tex]\[ P(x) = -0.002x^2 + 6.556x - 31 \][/tex]
Number of Containers to Maximize Profit:
[tex]\[ 1639 \][/tex]
Maximum Profit:
[tex]\[ 5341.642 \, \text{dollars} \][/tex]
Price to Charge Per Container:
[tex]\[ 3.478 \, \text{dollars} \][/tex]
These answers provide the detailed solution to the production and pricing problem for the small farm's ice cream containers.
