​(C) When each set is taxed at 

​$5​, 

the maximum profit is 

$

when 
 
sets are manufactured and sold for 

​$enter your response here 

each.

Let's complete the calculations for part (C) using the modified cost function that includes the $5 tax per set.

### Given:
- Original cost function: $ C(x) = 73,000 + 60x $
- Tax per set: $ $5 $

### New cost function:
\[ C_{new}(x) = 73,000 + (60 + 5)x \]
\[ C_{new}(x) = 73,000 + 65x \]

### New profit function:
\[ \Pi_{new}(x) = R(x) - C_{new}(x) \]
\[ \Pi_{new}(x) = \left(300x - \frac{x^2}{20}\right) - (73,000 + 65x) \]
\[ \Pi_{new}(x) = 300x - \frac{x^2}{20} - 73,000 - 65x \]
\[ \Pi_{new}(x) = 235x - \frac{x^2}{20} - 73,000 \]

To find the maximum profit, we differentiate $ \Pi_{new}(x) $ with respect to $ x $ and set the derivative to zero:
\[ \frac{d\Pi_{new}(x)}{dx} = 235 - \frac{x}{10} \]
\[ 235 - \frac{x}{10} = 0 \]
\[ x = 2350 \]

Now, we calculate the maximum profit:
\[ \Pi_{new}(2350) = 235(2350) - \frac{2350^2}{20} - 73,000 \]
\[ \Pi_{new}(2350) = 552,250 - 276,125 - 73,000 \]
\[ \Pi_{new}(2350) = 203,125 \]

The price per unit at maximum profit:
\[ p(2350) = 300 - \frac{2350}{20} \]
\[ p(2350) = 300 - 117.5 \]
\[ p(2350) = 182.5 \]

### Answer for part (C):
When each set is taxed at $5, the maximum profit is $203,125 when 2350 sets are manufactured and sold for $182.5 each.

Do you have any questions or need further details? Here are some other questions you might have:

1. How do you find the break-even point?
2. What is the impact of a price increase on profit?
3. How does a change in fixed costs affect the profit?
4. How can you derive the cost function from raw data?
5. What are some real-world applications of these types of problems?

**Tip:** Always verify your results by recalculating or using another method to confirm accuracy.

Learn how to maximize profit when each set is taxed at $5 using detailed calculations and analysis. This problem involves cost functions, profit maximization techniques, and differentiation methods. Suitable for advanced mathematics enthusiasts and students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation