We are given the following functions in the problem:

1. The revenue, $r(t)$, is modeled by: $r(t) = 900e^{0.3t}$

2. The number of customers, $c(t)$, is modeled by: $c(t) = \sqrt{200t}$

3. The revenue per customer, $g(t)$, is given by: $g(t) = \frac{r(t)}{c(t)}$

We are tasked with finding how fast the revenue earned per customer is changing on the 10th day, i.e., calculating $g'(10)$.

Step-by-step approach:

1. Differentiate $g(t)$: To find $g'(t)$, we need to use the quotient rule for derivatives: $\left( \frac{f(t)}{h(t)} \right)' = \frac{f'(t)h(t) - f(t)h'(t)}{h(t)^2}$ In this case: $f(t) = r(t) \quad \text{and} \quad h(t) = c(t)$

2. Differentiate $r(t)$: $r'(t) = \frac{d}{dt} \left( 900e^{0.3t} \right) = 900 \cdot 0.3e^{0.3t} = 270e^{0.3t}$

3. Differentiate $c(t)$: $c(t) = \sqrt{200t} = (200t)^{1/2}$ Using the chain rule: $c'(t) = \frac{1}{2}(200t)^{-1/2} \cdot 200 = \frac{100}{\sqrt{200t}}$

4. Apply the quotient rule: Now apply the quotient rule to find $g'(t)$: $g'(t) = \frac{r'(t) \cdot c(t) - r(t) \cdot c'(t)}{c(t)^2}$

   Substituting the expressions for $r'(t)$, $r(t)$, $c(t)$, and $c'(t)$: $g'(t) = \frac{(270e^{0.3t}) \cdot (\sqrt{200t}) - (900e^{0.3t}) \cdot \left( \frac{100}{\sqrt{200t}} \right)}{(\sqrt{200t})^2}$

   Simplify the expression step by step.

5. Evaluate at $t = 10$: Substitute $t = 10$ into the simplified expression to find $g'(10)$.

Let me now compute this.The rate at which the revenue earned per customer is changing on the 10th day is approximately 101.05 dollars per customer per day.

Would you like more details on any part of this solution?

Here are 5 related questions to expand on this topic:

1. How would the result change if the growth rate of revenue, $0.3$, was increased?
2. What happens to the revenue per customer if the number of customers grows at a faster rate?
3. How does the quotient rule help in calculating the derivative of revenue per customer?
4. What is the significance of the exponential function in the revenue model?
5. How can you interpret the second derivative $g''(t)$ in this context?

Tip: The quotient rule is essential for calculating the derivative of functions that are ratios, like revenue per customer.