https://file.aiquickdraw.com/mathbot/file/17308227565282y8fqr1k.jpg

To solve this problem, we are given the revenue function for the manufacturer:

\[
R(x) = 80x - 0.4x^2
\]

where $ x $ represents the number of units sold, and $ R(x) $ is the revenue in dollars. We need to find the maximum revenue and the number of units $ x $ that maximizes this revenue.

### Solution Steps:

1. **Find the derivative of $ R(x) $:**
   To find the maximum, we need to take the derivative of $ R(x) $ with respect to $ x $ and set it equal to zero.
   \[
   R'(x) = 80 - 0.8x
   \]

2. **Set the derivative to zero and solve for $ x $:**
   \[
   80 - 0.8x = 0
   \]
   \[
   0.8x = 80
   \]
   \[
   x = \frac{80}{0.8} = 100
   \]

   So, $ x = 100 $ units should be manufactured to achieve maximum revenue.

3. **Calculate the maximum revenue by substituting $ x = 100 $ back into $ R(x) $:**
   \[
   R(100) = 80(100) - 0.4(100)^2
   \]
   \[
   R(100) = 8000 - 0.4 \times 10000
   \]
   \[
   R(100) = 8000 - 4000 = 4000
   \]

### Answer:
- **Maximum Revenue:** $4000
- **Number of Units for Maximum Revenue:** 100 units

Would you like more details on any part of this solution, or do you have any other questions?

Here are some related questions to expand on this topic:

1. What is the significance of the second derivative in verifying maximum revenue?
2. How would the maximum revenue change if the function had a different quadratic term?
3. What are other methods to find maximum points of a quadratic function besides differentiation?
4. How does the concept of maximum revenue relate to the real-world pricing strategy?
5. How would we interpret the result if the derivative did not equal zero?

**Tip:** For optimization problems, always check the sign of the second derivative to confirm if the critical point is a maximum or a minimum.

A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function, R(x) = 80x - 0.4x^2 where the revenue is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum? (Round to nearest whole amount)

Discover how to find the maximum revenue and the number of units to produce for maximum revenue using the function R(x) = 80x - 0.4x^2. This problem involves calculus concepts such as optimization and the first derivative test, suitable for high school and college calculus students.

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