To solve the problem of finding the number of thousand MP3 players that should be produced to minimize the marginal cost and the minimum marginal cost itself, we need to follow these steps:
### Part a: Find the number of thousand MP3 Players to minimize the marginal cost.
1. Define the Marginal Cost Function:
[tex]\[
C(x) = x^2 - 160x + 7500
\][/tex]
2. Find the Critical Points by Taking the Derivative:
To find the number of units that minimize the marginal cost, we need to find the derivative of [tex]\( C(x) \)[/tex] and set it equal to zero. This will help us find the critical points.
[tex]\[
C'(x) = \frac{d}{dx}(x^2 - 160x + 7500)
\][/tex]
[tex]\[
C'(x) = 2x - 160
\][/tex]
3. Set the Derivative Equal to Zero:
[tex]\[
2x - 160 = 0
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
2x = 160
\][/tex]
[tex]\[
x = 80
\][/tex]
Therefore, the number of thousand MP3 players that should be produced to minimize the marginal cost is [tex]\( 80 \)[/tex] thousand.
### Part b: Find the Minimum Marginal Cost.
1. Evaluate the Marginal Cost Function at the Critical Point:
To find the minimum marginal cost, we substitute [tex]\( x = 80 \)[/tex] back into the original cost function [tex]\( C(x) \)[/tex]:
[tex]\[
C(80) = (80)^2 - 160(80) + 7500
\][/tex]
2. Calculate the Expression:
[tex]\[
C(80) = 6400 - 12800 + 7500
\][/tex]
[tex]\[
C(80) = 2625
\][/tex]
Therefore, the minimum marginal cost is [tex]\( 2625 \)[/tex] dollars.
### Summary:
- The number of thousand MP3 players that should be produced to minimize the marginal cost is [tex]\( 80 \)[/tex] thousand.
- The minimum marginal cost is [tex]\( 2625 \)[/tex] dollars.
