Question: The marginal cost of a product can be thought as the cost
of producing one additional unit of output. For example, if
the marginal cost of producing the 50th product is $6.20, 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 - 140x + 8000.
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, [blank] thousand mp3 players
should be produced.
1. **State the problem:** We are given the marginal cost function $$C(x) = x^2 - 140x + 8000$$ where $x$ is in thousands of mp3 players. We want to find:
A. The value of $x$ that minimizes the marginal cost.
B. The minimum marginal cost value.
2. **Recall the formula for the vertex of a parabola:** Since $C(x)$ is a quadratic function with a positive leading coefficient ($1$), it opens upwards and has a minimum at its vertex.
The vertex $x$-coordinate is given by $$x = -\frac{b}{2a}$$ where $a=1$ and $b=-140$.
3. **Calculate the vertex $x$-coordinate:**
$$x = -\frac{-140}{2 \times 1} = \frac{140}{2} = 70$$
So, the marginal cost is minimized when $x = 70$ thousand mp3 players.
4. **Find the minimum marginal cost by evaluating $C(70)$:**
$$C(70) = 70^2 - 140 \times 70 + 8000$$
$$= 4900 - 9800 + 8000$$
$$= (4900 + 8000) - 9800 = 12900 - 9800 = 3100$$
5. **Answer:**
A. To minimize the marginal cost, **70** thousand mp3 players should be produced.
B. The minimum marginal cost is **3100** dollars.