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?

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.