To find the solutions to the given problem, follow these steps:
### Step 1: Define Profit Function
The profit function, [tex]\(P(x)\)[/tex], is given by the difference between the Revenue [tex]\(R(x)\)[/tex] and the Cost [tex]\(C(x)\)[/tex].
1. Revenue, [tex]\(R(x)\)[/tex]: The revenue is calculated as the number of vats sold multiplied by the price per vat.
[tex]\[
R(x) = x \cdot p(x) = x \cdot (108 - 4x) = 108x - 4x^2
\][/tex]
2. Cost, [tex]\(C(x)\)[/tex]: The cost function is given as:
[tex]\[
C(x) = 4x + 2
\][/tex]
3. Profit, [tex]\(P(x)\)[/tex]: The profit is the revenue minus the cost:
[tex]\[
P(x) = R(x) - C(x) = (108x - 4x^2) - (4x + 2)
\][/tex]
Simplifying this, we get:
[tex]\[
P(x) = -4x^2 + 104x - 2
\][/tex]
### Step 2: Find the Number of Vats to Maximize Profit
To find the number of vats, [tex]\(x\)[/tex], that maximizes the profit, we need to find the critical points of [tex]\(P(x)\)[/tex]. This is done by taking the derivative of [tex]\(P(x)\)[/tex] with respect to [tex]\(x\)[/tex] and setting it to zero.
1. Derivative, [tex]\(P'(x)\)[/tex]:
[tex]\[
P'(x) = \frac{d}{dx}(-4x^2 + 104x - 2) = -8x + 104
\][/tex]
Setting the derivative equal to zero to find the critical points:
[tex]\[
-8x + 104 = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{104}{8} = 13
\][/tex]
So, 13 vats of ice cream need to be sold to maximize the profit.
### Step 3: Find the Maximum Profit
To find the maximum profit, substitute [tex]\(x = 13\)[/tex] into the profit function [tex]\(P(x)\)[/tex].
[tex]\[
P(13) = -4(13)^2 + 104(13) - 2
\][/tex]
Calculating this, we get:
[tex]\[
P(13) = -4(169) + 1352 - 2 = -676 + 1352 - 2 = 674
\][/tex]
Therefore, the maximum profit is [tex]$674.
### Step 4: Find the Price to Charge Per Vat
To find the price to charge per vat to maximize the profit, substitute \(x = 13\) into the price-demand function \(p(x)\):
\[
p(13) = 108 - 4(13)
\]
Calculating this, we get:
\[
p(13) = 108 - 52 = 56
\]
So, the price to charge per vat to maximize profit is $[/tex]56.
### Summary
- [tex]\(\boxed{P(x) = -4x^2 + 104x - 2}\)[/tex]
- 13 vats of ice cream need to be sold to maximize the profit. [tex]\(\boxed{13}\)[/tex]
- The maximum profit is [tex]$674. \(\boxed{674}\)
- The price to charge per vat to maximize the profit is $[/tex]56. [tex]\(\boxed{56}\)[/tex]
