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To answer this problem, let's go through each part step-by-step.

### Given Information
The revenue $ R(x) $ in thousands of dollars is provided by the formula:
\[
R(x) = x(30 - 3x)
\]
where $ x $ represents the number of thousand compact disc (CD) players produced.

---

### Part (a): Evaluate $ R(2) $
To find $ R(2) $, substitute $ x = 2 $ into the revenue function.

\[
R(2) = 2(30 - 3 \cdot 2)
\]
\[
= 2(30 - 6)
\]
\[
= 2 \times 24 = 48
\]

So, $ R(2) = 48 $. 

**Interpretation**: The revenue generated by producing 2,000 CD players is 48 thousand dollars, or $48,000.

---

### Part (b): Determine the Number of CD Players to Maximize Revenue
To maximize the revenue, we need to find the vertex of the parabola represented by $ R(x) = x(30 - 3x) $.

1. Rewrite the revenue function in standard quadratic form:
   \[
   R(x) = 30x - 3x^2
   \]

2. The general form of a quadratic equation is $ ax^2 + bx + c $. In this case:
   - $ a = -3 $
   - $ b = 30 $

3. For a quadratic function $ ax^2 + bx + c $, the maximum (vertex) occurs at $ x = -\frac{b}{2a} $:
   \[
   x = -\frac{30}{2(-3)} = \frac{30}{6} = 5
   \]

So, to maximize revenue, the company should produce 5,000 CD players.

---

### Part (c): Calculate the Maximum Revenue
Substitute $ x = 5 $ into the revenue function to find the maximum revenue.

\[
R(5) = 5(30 - 3 \cdot 5)
\]
\[
= 5(30 - 15)
\]
\[
= 5 \times 15 = 75
\]

**Maximum Revenue**: The maximum revenue is 75 thousand dollars, or $75,000.

---

Would you like more details on any of these calculations or steps?

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#### Follow-up Questions
1. What happens to the revenue if the production increases beyond 5,000 CD players?
2. How would the formula change if the company introduced a cost per unit for production?
3. How is the maximum revenue affected if the demand increases by a certain percentage?
4. What other factors could influence the production number needed for maximum revenue?
5. Can you derive a general formula for the revenue-maximizing production level if the coefficients change?

#### Tip
For any quadratic revenue function in the form $ ax^2 + bx + c $, finding the vertex gives the production level that maximizes revenue.

Suppose the revenue R in thousands of dollars that a company receives from producing x thousand compact disc players is given by the formula R(x) = x(30 – 3x). a. Evaluate R(2) and interpret the result. b. How many CD players should the company produce to maximize its revenue? c. What is the maximum revenue?

Learn how to evaluate and maximize revenue for a company producing CD players using quadratic functions. Solve for R(2), find the optimal production level, and calculate the maximum revenue step-by-step. Suitable for students in Grades 10-12.

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