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!