The profit function is given by:

$P(n) = 8n - \sqrt{n}$

where $n$ is the number of subscribers. You're asked to find:

1. The current profit when $n = 100$ subscribers.
2. The marginal profit, which is the derivative of the profit function $P(n)$ with respect to $n$, evaluated at $n = 100$.

Step 1: Calculate the profit at $n = 100$

We substitute $n = 100$ into the profit function:

$P(100) = 8(100) - \sqrt{100} = 800 - 10 = 790$

So, your current profit is $790 per month.

Step 2: Find the marginal profit

To find the marginal profit, we first take the derivative of the profit function $P(n)$:

$P'(n) = \frac{d}{dn}(8n - \sqrt{n}) = 8 - \frac{1}{2\sqrt{n}}$

Now, evaluate this derivative at $n = 100$:

$P'(100) = 8 - \frac{1}{2\sqrt{100}} = 8 - \frac{1}{20} = 8 - 0.05 = 7.95$

Thus, the marginal profit is $7.95 per new subscriber.

Interpretation

Your current profit is $790 per month, and for each additional subscriber, your profit will increase by approximately $7.95. This means that adding new subscribers has a slightly diminishing impact on profit growth due to the square root term.

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Do you need further details or have any questions?
Here are some related questions to explore further:

1. What happens to the profit as the number of subscribers grows larger?
2. How does the marginal profit behave as the number of subscribers increases?
3. Can the marginal profit ever be negative?
4. What is the maximum possible profit you can achieve with this model?
5. How would the profit change if the equation included a fixed cost per subscriber?

Tip: The marginal profit helps determine how efficiently you're increasing your earnings with each new subscriber!