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For a small farm, the cost in dollars to produce [tex]x[/tex] containers of ice cream is [tex]C(x) = 0.2x + 31[/tex]. The price-demand function, in dollars, is [tex]p(x) = 6.756 - 0.002x[/tex].

1. Find the profit function:
[tex]\[
P(x) = \square
\][/tex]

2. How many containers of ice cream need to be sold to maximize the profit?
[tex]\[
\square \quad \text{(Select an answer)}
\][/tex]

3. Find the maximum profit:
[tex]\[
\square \quad \text{(Select an answer)}
\][/tex]

4. Find the price to charge per container to maximize profit:
[tex]\[
\square \quad \text{(Select an answer)}
\][/tex]

Sagot :

GinnyAnsweridn
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.
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