To find how fast the revenue earned per customer is changing on the 9th day, we need to calculate the rate of change of $g(t) = \frac{r(t)}{c(t)}$ with respect to time, $t$, and then evaluate it at $t = 9$.

Step 1: Define the functions

We have the following functions:

• Revenue function: $r(t) = 900e^{0.31t}$
• Customer function: $c(t) = \sqrt{240t} = (240t)^{1/2}$
• Revenue per customer: $g(t) = \frac{r(t)}{c(t)}$

Step 2: Differentiate $g(t)$

We use the quotient rule to differentiate $g(t)$: [ g'(t) = \frac{ \frac{d}{dt}[r(t)] \cdot c(t) - r(t) \cdot \frac{d}{dt}[c(t)] }{[c(t)]^2} ]

First, find the derivatives of $r(t)$ and $c(t)$.

Derivative of $r(t)$:

$r'(t) = \frac{d}{dt}[900e^{0.31t}] = 900 \cdot 0.31e^{0.31t} = 279e^{0.31t}$

Derivative of $c(t)$:

$c'(t) = \frac{d}{dt}[(240t)^{1/2}] = \frac{1}{2}(240t)^{-1/2} \cdot 240 = \frac{120}{\sqrt{240t}}$

Step 3: Plug these into the quotient rule

Now substitute into the quotient rule: $g'(t) = \frac{279e^{0.31t} \cdot \sqrt{240t} - 900e^{0.31t} \cdot \frac{120}{\sqrt{240t}}}{(240t)}$

Step 4: Evaluate at $t = 9$

We need to calculate $g'(9)$. First, plug $t = 9$ into the expressions for $r(t)$, $r'(t)$, $c(t)$, and $c'(t)$.

• $r(9) = 900e^{0.31 \times 9} = 900e^{2.79}$
• $r'(9) = 279e^{2.79}$
• $c(9) = \sqrt{240 \times 9} = \sqrt{2160}$
• $c'(9) = \frac{120}{\sqrt{2160}}$

Now, substitute these values into the expression for $g'(9)$ and simplify.

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This will give the rate at which the revenue per customer is changing on the 9th day.

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