Answer:
The least number of items to produce is 41
Step-by-step explanation:
Average Cost
Given C(x) as the cost function to produce x items. The average cost is:
[tex]\displaystyle \bar C(X)=\frac{C(x)}{x}[/tex]
The cost function is:
[tex]C(x) = -20x+1681[/tex]
And the average cost function is:
[tex]\displaystyle \bar C(X)=\frac{-20x+1681}{x}[/tex]
We are required to find the least number of items that can be produced so the average cost is less or equal to $21.
We set the inequality:
[tex]\displaystyle \frac{-20x+1681}{x}\le 21[/tex]
Multiplying by x:
[tex]-20x+1681 \le 21x[/tex]
Note we multiplied by x and did not flip the inequality sign because its value cannot be negative.
Adding 20x:
[tex]1681 \le 21x+20x[/tex]
[tex]1681 \le 41x[/tex]
Swapping sides and changing the sign:
[tex]41x \ge 1681[/tex]
Dividing by 41:
[tex]x\ge 41[/tex]
The least number of items to produce is 41
