Suppose the revenue $$ R $$ in thousands of dollars that a company receives from producing $$ x $$ thousand compact disc players is given by the formula $$ R(x)=x(50-5 x) $$. a. Evaluate $$ R(3) $$ and interpret the result. b. How many CD players should the company produce to maximize its revenue? c. What is the maximum revenue? a. Evaluate $$ R(3) $$. $$ R(3)=105 $$ Interpret the result. Choose the correct answer below. The company receives $$ \$ 105,000 $$ for producing 3000 CD players. The company gives $$ \$ 105,000 $$ for purchasing 3000 CD players.

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**a. Evaluating $$ R(3) $$**

We substitute $$ x = 3 $$ into the revenue function:
$$
R(3)=3(50-5\cdot 3)=3(50-15)=3(35)=105.
$$
Since $$ R(x) $$ is measured in thousands of dollars and $$ x $$ in thousands of CD players, $$ R(3)=105 $$ means the company receives revenue of $$ 105 $$ thousand dollars, i.e., \$105,000 for producing 3000 CD players.

**b. Maximizing the Revenue**

The revenue function is:
$$
R(x)=x(50-5x)=50x-5x^2.
$$
This is a quadratic function in the form $$ R(x)=ax^2+bx+c $$ with $$ a=-5 $$ and $$ b=50 $$. The maximum revenue occurs at the vertex of the parabola. The $$ x $$-coordinate of the vertex is given by:
$$
x=-\frac{b}{2a}=-\frac{50}{2(-5)}=\frac{50}{10}=5.
$$
Thus, the revenue is maximized when the company produces $$ 5 $$ thousand CD players (5000 CD players).

**c. Maximum Revenue**

Substitute $$ x=5 $$ into $$ R(x) $$:
$$
R(5)=5(50-5\cdot 5)=5(50-25)=5(25)=125.
$$
Since revenue is in thousands of dollars, $$ 125 $$ corresponds to \$125,000.

**Summary of Answers:**

- **a.** $$ R(3)=105 $$ and the company receives \$105,000 for producing 3000 CD players.
- **b.** The company should produce 5000 CD players to maximize revenue.
- **c.** The maximum revenue is \$125,000.
**a.** When the company produces 3000 CD players, it earns \$105,000. **b.** To maximize revenue, the company should produce 5000 CD players. **c.** The maximum revenue is \$125,000.

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