Let's solve the given problem step by step:

Part (a): Construct a linear demand function

We are given two data points for the price-demand relationship:

• When the log-on fee $x = 4$, the demand is $570$ log-ons per month.
• When the log-on fee $x = 3.50$, the demand is $855$ log-ons per month.

To construct the linear demand function, we can use the form of a linear equation: $D(x) = m x + b$ where $D(x)$ represents the demand at a log-on fee $x$, and $m$ and $b$ are constants.

Step 1: Find the slope ($m$)

The slope $m$ is given by: $m = \frac{\Delta D}{\Delta x} = \frac{855 - 570}{3.50 - 4} = \frac{285}{-0.5} = -570$

Step 2: Find the intercept ($b$)

We can use one of the points to find $b$. Let's use $(x, D) = (4, 570)$: $570 = -570(4) + b$ $570 = -2280 + b$ $b = 2850$

Thus, the demand function is: $D(x) = -570x + 2850$

The revenue function $R(x)$ is given by: $R(x) = x \cdot D(x) = x(-570x + 2850) = -570x^2 + 2850x$

Part (b): Monthly Profit Function

The monthly profit is the revenue minus the costs. We are given a fixed monthly cost of $50 to maintain the site.

Thus, the profit function $P(x)$ is: $P(x) = R(x) - 50 = -570x^2 + 2850x - 50$

Step 3: Find the maximum profit

To maximize the profit, we need to find the critical points of $P(x)$ by setting its derivative equal to zero: $P'(x) = -1140x + 2850$ Set $P'(x) = 0$: $-1140x + 2850 = 0$ $1140x = 2850$ $x = \frac{2850}{1140} = 2.5$

So, the log-on fee that maximizes profit is $2.50.

Step 4: Calculate the maximum profit

Substitute $x = 2.5$ into the profit function: $P(2.5) = -570(2.5)^2 + 2850(2.5) - 50$ $P(2.5) = -570(6.25) + 7125 - 50$ $P(2.5) = -3562.5 + 7125 - 50$ $P(2.5) = 3512.5$

The largest possible monthly profit is $3512.50.

Would you like more details or have any questions?

5 Related Questions:

1. How does changing the fixed monthly cost affect the maximum profit?
2. Can the demand function still be linear if additional data points do not fit the current trend?
3. What happens to the profit if the log-on fee is increased beyond $4?
4. How would the profit function change if there were additional variable costs per log-on?
5. How does calculus help in optimizing business models like this?

Tip:

When constructing linear functions, always use at least two data points to find the slope and y-intercept accurately.