Problem Breakdown

1. Profit Function Components:

   • The revenue per person starts at $180 (when 40 people attend) and decreases by $2 for each additional person over 40.
   • The total cost includes:
     • A fixed cost of $6000.
     • A variable cost of $28 per person.

2. Define Variables:

   • Let $x$ represent the total number of people attending the tour.
   • Revenue per person = $180 - 2(x - 40) = 260 - 2x$ (derived from the decreasing rate).
   • Total revenue = $x \cdot (260 - 2x) = 260x - 2x^2$.
   • Total cost = $6000 + 28x$.

3. Profit Function: Profit $f(x)$ = Total Revenue - Total Cost: $f(x) = (260x - 2x^2) - (6000 + 28x)$ Simplify: $f(x) = -2x^2 + 232x - 6000$

4. Maximize Profit: To maximize $f(x)$, we find the vertex of the parabola. The formula for the $x$-coordinate of the vertex is: $x = -\frac{b}{2a}$ where $a = -2$ and $b = 232$.

Calculations

$x = -\frac{232}{2(-2)} = \frac{232}{4} = 58$

Thus, the number of people that maximizes the profit is $x = 58$.

Final Answers

1. Profit function: $f(x) = -2x^2 + 232x - 6000$

2. Number of people that maximizes profit: $x = 58$

Would you like a deeper explanation of any step or how this applies to similar problems?

Related Questions

1. How is the revenue function derived when rates decrease linearly?
2. What is the significance of the vertex in a quadratic profit model?
3. How do fixed and variable costs affect the profit function?
4. Can the solution be checked using derivatives? How?
5. What happens to the profit if fewer than 40 or more than 80 people attend?

Tip

To maximize profit in similar scenarios, always model your revenue and cost functions clearly, and remember that a quadratic function will have a single maximum if the leading coefficient is negative.