The marginal cost of a product can be thought of as the cost of producing one additional unit of output. For example, if the marginal cost of producing the 50 th product is $$ \$ 620 $$, it cost $$ \$ 6.20 $$ to increase production from 49 to 50 units of output. Suppose the marginal cost $$ C $$ (in dollars) to produce $$ x $$ thousand mp3 players is given by the function $$ C(x)=x^{2}-120 x+8400 $$. A. How many players should be produced to minimize the marginal cost? B. What is the minimum marginal cost? A. To minimize the marginal cost, $$ \square $$ thousand mp3 players should be produced. B. The minimum marginal cost is $$ \$ \square $$

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UpStudy AI · Accepted Answer

Let $$ C(x) = x^2 - 120x + 8400 $$ represent the marginal cost (in dollars) for producing $$ x $$ thousand mp3 players.

**Step 1. Identify the Vertex of the Parabola**

Since $$ C(x) $$ is a quadratic function in the form
$$
ax^2 + bx + c,
$$
the function reaches its minimum at the vertex when $$ a > 0 $$. The $$ x $$-coordinate of the vertex is given by
$$
x = -\frac{b}{2a}.
$$
For our function, $$ a = 1 $$ and $$ b = -120 $$. Thus,
$$
x = -\frac{-120}{2(1)} = \frac{120}{2} = 60.
$$

**Step 2. Interpret the Result**

Since $$ x $$ represents thousands of mp3 players, $$ 60 $$ means $$ 60 $$ thousand mp3 players.

**Step 3. Calculate the Minimum Marginal Cost**

Substitute $$ x = 60 $$ into $$ C(x) $$:
$$
C(60) = 60^2 - 120(60) + 8400.
$$
Computing step by step:
$$
60^2 = 3600,
$$
$$
120 \times 60 = 7200.
$$
Thus,
$$
C(60) = 3600 - 7200 + 8400.
$$
Combine the terms:
$$
3600 - 7200 = -3600,
$$
$$
-3600 + 8400 = 4800.
$$

**Final Answers:**

A. To minimize the marginal cost, $$ \boxed{60} $$ thousand mp3 players should be produced.  
B. The minimum marginal cost is $$ \$\boxed{4800} $$.
A. To minimize the marginal cost, 60 thousand mp3 players should be produced. B. The minimum marginal cost is \$4800.

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