A manufacturer incurs the following costs in producing $$ x $$ water ski vests in one day, for $$ 0

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UpStudy AI · Accepted Answer

**(A) Average Cost per Vest**

The cost function is  
$$
C(x)=0.05 x^2+20x+245
$$  
for $$ 00 $$, the only critical number in the domain is $$ x=70 $$.

3. **Determine the intervals of increase and decrease:**

Test a value $$ x $$ less than 70, e.g., $$ x=50 $$:
   $$
   \overline{C}'(50)=0.05-\frac{245}{50^2}=0.05-\frac{245}{2500}=0.05-0.098=-0.048.
   $$
   The derivative is negative, so $$\overline{C}(x)$$ is decreasing on $$(0,70)$$.

Test a value $$ x $$ greater than 70, e.g., $$ x=100 $$:
   $$
   \overline{C}'(100)=0.05-\frac{245}{100^2}=0.05-\frac{245}{10000}=0.05-0.0245=0.0255.
   $$
   The derivative is positive, so $$\overline{C}(x)$$ is increasing on $$(70,175)$$.

4. **Local Extremum:**

Since the function changes from decreasing to increasing at $$ x=70 $$, there is a local minimum at $$ x=70 $$.

5. **Evaluate the Minimum Average Cost:**

$$
   \overline{C}(70)=0.05(70)+20+\frac{245}{70}=3.5+20+3.5=27.
   $$

---

**Conclusion:**

The critical number is $$ x=70 $$. The average cost per vest is decreasing on $$(0,70)$$ and increasing on $$(70,175)$$. There is a local minimum at $$ x=70 $$ with  
$$
\overline{C}(70)=27.
$$
Thus, the correct choice is:  
**A. $$ \overline{C}(70)=27 $$ is a local minimum. There is no local maximum.**
The average cost per vest is minimized at 27 when producing 70 vests.

A manufacturer incurs the following costs in producing \( x \) water ski vests in one day, for \( 0

Ask by Klein Owen. in the United States

Mar 12,2025

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A manufacturer incurs the following costs in producing \( x \) water ski vests in one day, for \( 0 Ask by Klein Owen. in the United States Mar 12,2025