You operate a gaming website, www.mudbeast.net, where users must pay a small fee to log on. When you charged $4 the demand was 520 log-ons per month. When you lowered the price to $3.50, the demand increased to 780 log-ons per month.
(a)
Construct a linear demand function for your website and hence obtain the monthly revenue R as a function of the log-on fee x.
R(x) =

(b)
Your Internet provider charges you a monthly fee of $20 to maintain your site. Express your monthly profit P as a function of the log-on fee x.
P(x) =
Determine the log-on fee you should charge to obtain the largest possible monthly profit (in dollars).
x = $
What is the largest possible monthly profit (in dollars)?

### (a) Constructing a Linear Demand Function

The problem provides two points from the demand function, where:

- At a price of $ x = 4 $, the demand (log-ons per month) is $ 520 $,
- At a price of $ x = 3.5 $, the demand is $ 780 $.

We can assume the demand function is linear and can be written in the form:

\[
D(x) = mx + b
\]

where $ D(x) $ is the demand (log-ons per month) and $ x $ is the log-on fee. To find $ m $ (the slope), we use the two points $ (4, 520) $ and $ (3.5, 780) $:

\[
m = \frac{780 - 520}{3.5 - 4} = \frac{260}{-0.5} = -520
\]

So the slope $ m = -520 $.

Now, we use the point-slope form of the line to find $ b $. Substituting one of the points, say $ (4, 520) $, into the equation $ D(x) = -520x + b $:

\[
520 = -520(4) + b
\]
\[
520 = -2080 + b
\]
\[
b = 2600
\]

Thus, the demand function is:

\[
D(x) = -520x + 2600
\]

### Revenue Function $ R(x) $

The revenue is the product of the log-on fee $ x $ and the demand $ D(x) $:

\[
R(x) = x \cdot D(x) = x \cdot (-520x + 2600)
\]
\[
R(x) = -520x^2 + 2600x
\]

### (b) Profit Function $ P(x) $

The profit function is the revenue minus the cost. The only cost mentioned is the monthly fee of $20 charged by the Internet provider. Thus, the profit function is:

\[
P(x) = R(x) - 20
\]
\[
P(x) = (-520x^2 + 2600x) - 20
\]
\[
P(x) = -520x^2 + 2600x - 20
\]

### Maximizing the Profit

To find the log-on fee $ x $ that maximizes profit, we take the derivative of the profit function $ P(x) $ and set it equal to zero:

\[
P'(x) = \frac{d}{dx}(-520x^2 + 2600x - 20)
\]
\[
P'(x) = -1040x + 2600
\]

Setting $ P'(x) = 0 $:

\[
-1040x + 2600 = 0
\]
\[
1040x = 2600
\]
\[
x = \frac{2600}{1040} = 2.5
\]

So, the log-on fee that maximizes the profit is $ x = 2.5 $.

### Largest Possible Monthly Profit

To find the largest possible monthly profit, substitute $ x = 2.5 $ into the profit function $ P(x) $:

\[
P(2.5) = -520(2.5)^2 + 2600(2.5) - 20
\]
\[
P(2.5) = -520(6.25) + 6500 - 20
\]
\[
P(2.5) = -3250 + 6500 - 20
\]
\[
P(2.5) = 3230
\]

Thus, the largest possible monthly profit is **$3230**.

---

### Summary:
- The demand function is $ D(x) = -520x + 2600 $.
- The revenue function is $ R(x) = -520x^2 + 2600x $.
- The profit function is $ P(x) = -520x^2 + 2600x - 20 $.
- The log-on fee that maximizes profit is $ x = 2.5 $ dollars.
- The largest possible monthly profit is **$3230**.

---

Do you want more details on any part of the solution?

Here are 5 related questions:
1. What does the slope of the demand function represent in this context?
2. How would the profit change if the monthly site maintenance fee increased?
3. What is the price elasticity of demand based on the given data points?
4. How does maximizing revenue differ from maximizing profit in this scenario?
5. How would the optimal log-on fee change if the demand function were quadratic instead of linear?

**Tip**: When finding maximum or minimum values in quadratic functions, the vertex formula $ x = \frac{-b}{2a} $ is a quick and efficient method.

You operate a gaming website, www.mudbeast.net, where users must pay a small fee to log on. When you charged $4 the demand was 520 log-ons per month. When you lowered the price to $3.50, the demand increased to 780 log-ons per month.
(a) Construct a linear demand function for your website and hence obtain the monthly revenue R as a function of the log-on fee x.
(b) Your Internet provider charges you a monthly fee of $20 to maintain your site. Express your monthly profit P as a function of the log-on fee x. Determine the log-on fee you should charge to obtain the largest possible monthly profit (in dollars).
What is the largest possible monthly profit (in dollars)?

This problem demonstrates how to construct a linear demand function, calculate revenue, and express monthly profit for a gaming website. Learn how to find the optimal log-on fee for maximum profit using calculus-based optimization. Suitable for high school and early college students.

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