Your monthly profit (in dollars) from your newspaper route is given by

*P* = 8*n* − <square root>n,

where *n* is the number of subscribers on your route. If you currently have 100 subscribers, find your profit and your marginal profit.

profit$ marginal profit$  per new subscriber

Interpret your answers.

Your current profit is $  per month and this would increase at a rate of $  per new subscriber.




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.

---

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!

Your monthly profit (in dollars) from your newspaper route is given by P = 8n − √n, where n is the number of subscribers on your route. If you currently have 100 subscribers, find your profit and your marginal profit. Interpret your answers.

Calculate your current profit and marginal profit based on the function P = 8n − √n, where n represents subscribers. Learn how profit increases at a rate of $7.95 per new subscriber when you have 100 subscribers. Suitable for high school students learning introductory calculus concepts.

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