Q. 89 - Precalculus Enhanced with Graphing Utilities

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

$f\left(x\right)=x^2-140x+7400$

(a) How many players should be produced to minimize the marginal cost?

(b) What is the minimum marginal cost?

Step-by-Step Solution

Answer

a) The number of players should be produced is $70$

b) the minimum marginal cost is $2500

Part a) Step 1: Given information

The given equation is $C\left(x\right)=x^2-140x+7400$

Part a) Step 2: Find the vertex

The coefficient of $x^2$is positive. hence the parabola is open upward and the minimum is reached at vertex

The vertex can be given as

$$\begin{gathered} x=-\frac{b}{2a} \\ x=-\frac{-140}{2} \\ x=70 \end{gathered}$$

Hence 70 thousand digital music players should be produced to minimize the marginal cost

Part b) Step 1: TO find the minimum marginal cost substitute x = 70 in the equation

We get,

$$\begin{gathered} f\left(x\right)=x^2-140x+7400 \\ f\left(x\right)={70}^2-\left(140\right)\left(70\right)+7400 \\ f\left(x\right)=4900+9800+7400 \\ f\left(x\right)=2500 \end{gathered}$$

the minimum marginal cost is $2500

Step 4: Conclusion

a) The number of players should be produced is 70 thousand

b) the minimum marginal cost is $2500