Question

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 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)=x² - 160x + 8100.
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, thousand mp3 players should be produced.

Explanation:

Step1: Identify the function type

The marginal - cost function $C(x)=x^{2}-160x + 8100$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a = 1$, $b=-160$, and $c = 8100$.

Step2: Find the x - value of the vertex

For a quadratic function $y = ax^{2}+bx + c$, the x - coordinate of the vertex (which gives the minimum value when $a>0$) is $x=-\frac{b}{2a}$. Substitute $a = 1$ and $b=-160$ into the formula: $x=-\frac{-160}{2\times1}=\frac{160}{2}=80$.

Step3: Find the minimum value of the function

Substitute $x = 80$ into the cost function $C(x)=x^{2}-160x + 8100$.
\[

$$\begin{align*} C(80)&=(80)^{2}-160\times80 + 8100\\ &=6400-12800 + 8100\\ &=6400+8100-12800\\ &=14500 - 12800\\ &=1700 \end{align*}$$

\]

Answer:

A. 80
B. $1700$