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We are given that the revenue $ R(x) $ from selling $ x $ units of an item is given by the equation:

\[
R(x) = 1200x - 2x^2
\]

We are also told that sales are increasing at the rate of 60 units per day, i.e., $ \frac{dx}{dt} = 60 $ units per day. We need to determine how rapidly the revenue is increasing, i.e., $ \frac{dR}{dt} $, when 200 units have been sold (i.e., $ x = 200 $).

### Steps:
1. **Differentiate the revenue function with respect to time**:  
   We can use the chain rule to differentiate $ R(x) $ with respect to time $ t $:
   \[
   \frac{dR}{dt} = \frac{dR}{dx} \cdot \frac{dx}{dt}
   \]
   
2. **Find $ \frac{dR}{dx} $**:  
   Differentiate $ R(x) = 1200x - 2x^2 $ with respect to $ x $:
   \[
   \frac{dR}{dx} = 1200 - 4x
   \]

3. **Evaluate $ \frac{dR}{dx} $ at $ x = 200 $**:
   \[
   \frac{dR}{dx} = 1200 - 4(200) = 1200 - 800 = 400
   \]

4. **Substitute $ \frac{dx}{dt} = 60 $ and $ \frac{dR}{dx} = 400 $ into the chain rule equation**:
   \[
   \frac{dR}{dt} = 400 \cdot 60 = 24,000
   \]

So, the revenue is increasing at a rate of **$24,000 per day** when 200 units have been sold.

---

Would you like further details on any of the steps? Here are 5 related questions to deepen your understanding:

1. What happens to the rate of revenue increase if the sales rate $ \frac{dx}{dt} $ changes?
2. How would the result change if the revenue function had a different quadratic term?
3. Can you calculate the maximum possible revenue based on the given formula?
4. What does the second derivative of $ R(x) $ tell us about the revenue function?
5. How does the relationship between $ x $ and $ R(x) $ change as $ x $ becomes very large?

**Tip**: Understanding how to apply the chain rule in related rates problems is crucial, especially when dealing with variables that change with time!

A company's revenue from selling x units of an item is given as R = 1200x - 2x^2. If sales are increasing at the rate of 60 units per day, how rapidly is revenue increasing (in dollars per day) when 200 units have been sold?

Learn how to calculate the rate of change in revenue when sales are increasing at 60 units per day, given the revenue function R = 1200x - 2x^2. This calculus problem demonstrates the application of related rates and differentiation. Suitable for students in Grades 11-12.

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