Chapter 4.6, Problem 77AYU

Average Cost Suppose that the daily cost $C$ of manufacturing bicycles is given by $C(x)=80x\text{ + 5000}$. Then the average daily cost $\bar{C}$ is given by $\bar{C}(x)=\frac{80x\text{ + 5000}}{x}$. How many bicycles must be produced each day for the average cost to be no more than $\$100$?

To determine

To find: How many bicycles must be produced each day for the average cost to be no more than $\$100?$

Answer to Problem 77AYU

At least 250.

Explanation of Solution

Given:

$C\left(x\right) = 80x + 5000$ and $\overline{C\left(x\right)} = \frac{80x + 5000}{x}$

Average cost to be no more than $\$100$.

$\frac{80x + 5000}{x} \le 100$

$\frac{80x + 5000}{x}-100 \le 0$

$\frac{5000 - 20x}{x} \le 0$

The zero of $F\left(x\right) = \frac{5000 - 20x}{x}$ is $x = 250$.

Use the zero and the undefined value to separate the real number line into intervals:

$\left(-\infty , 250\right], \left[250, \infty \right)$

Select a test number in each interval found in above and evaluate $F\left(x\right) = \frac{5000 - 20x}{x}$ each number to determine if $F\left(x\right)$ is positive or negative.

Interval	$\left(-\infty ,250\right]$	$\left[250, \infty \right)$
Number chosen	200	300
Value of $f$	5	$-\frac{10}{3}$
Conclusion	Positive	Negative

Since we want to know where $F\left(x\right)$ is negative, conclude that $F\left(x\right) \le 0$. The solution set is $\left\{x|250 \le x < \infty \right\}$ or, using interval notation $\left[250, \infty \right)$.

Therefore at least 250 bicycles must be produced each day for the average cost to be no more than $\$100$.