Let's break down the problem and solve each part step by step.

Part (a): Find the simplified cost function $C(x)$.

The cost function $C(x)$ is the sum of the fixed costs and the variable costs. The fixed cost is given as $45,000. The variable cost per unit is $\frac{4}{9}x + 222$ dollars.

Thus, the total cost function $C(x)$ is: $C(x) = \text{Fixed Costs} + \text{Variable Costs}$ $C(x) = 45000 + x \times \left( \frac{4}{9}x + 222 \right)$

Simplifying the expression: $C(x) = 45000 + \left( \frac{4}{9}x^2 + 222x \right)$ $C(x) = \frac{4}{9}x^2 + 222x + 45000$

So, the simplified cost function is: $C(x) = \frac{4}{9}x^2 + 222x + 45000$

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Part (b): Find the simplified revenue function $R(x)$.

The revenue function $R(x)$ is the selling price per unit times the number of units produced, $x$. The selling price per unit is $1752 - \frac{5}{9}x$.

Thus, the revenue function $R(x)$ is: $R(x) = x \times \left( 1752 - \frac{5}{9}x \right)$ Simplifying the expression: $R(x) = 1752x - \frac{5}{9}x^2$

So, the simplified revenue function is: $R(x) = 1752x - \frac{5}{9}x^2$

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Part (c): Form the profit function $P(x)$.

The profit function $P(x)$ is the difference between the revenue and the cost functions: $P(x) = R(x) - C(x)$ Substitute the expressions for $R(x)$ and $C(x)$: $P(x) = \left( 1752x - \frac{5}{9}x^2 \right) - \left( \frac{4}{9}x^2 + 222x + 45000 \right)$ Simplifying: $P(x) = 1752x - \frac{5}{9}x^2 - \frac{4}{9}x^2 - 222x - 45000$ $P(x) = 1752x - 222x - \left( \frac{5}{9}x^2 + \frac{4}{9}x^2 \right) - 45000$ $P(x) = 1530x - \frac{9}{9}x^2 - 45000$ $P(x) = 1530x - x^2 - 45000$

Thus, the simplified profit function is: $P(x) = 1530x - x^2 - 45000$

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Would you like more details on any of these steps or have any specific questions? Here are some related questions:

1. How do fixed and variable costs impact the cost function?
2. What happens to the revenue if the number of units produced increases significantly?
3. How can the company maximize its profit given the profit function $P(x)$?
4. What does the quadratic term in the profit function suggest about the relationship between production and profit?
5. How would changing the fixed costs affect the cost function?

Tip: Maximizing the profit function can be done by finding the derivative of $P(x)$ and setting it equal to zero to solve for $x$. This will give the number of units that maximize profit.