Let's break this down step by step:

Step 1: Expression for Total Revenue $R(x)$

• The plane holds 100 passengers, and there are $x$ unsold seats.
• This means the number of passengers on the flight is $100 - x$.
• The fare per person is $300$, but there is an additional charge of $40$ for each unsold seat.

Thus, the fare per person (accounting for unsold seats) is:

$\text{Fare per person} = 300 + 40x$

Now, the total number of paying passengers is $100 - x$, so the total revenue $R(x)$ can be written as:

$R(x) = (100 - x) \times (300 + 40x)$

Step 2: Expand the Expression for $R(x)$

Now, expand the equation to simplify it:

$R(x) = (100 - x)(300 + 40x)$ $R(x) = 100(300 + 40x) - x(300 + 40x)$ $R(x) = 30{,}000 + 4{,}000x - 300x - 40x^2$ $R(x) = 30{,}000 + 3{,}700x - 40x^2$

So, the expression for total revenue is:

$R(x) = -40x^2 + 3{,}700x + 30{,}000$

Step 3: Find the Number of Unsold Seats for Maximum Revenue

The revenue function $R(x) = -40x^2 + 3{,}700x + 30{,}000$ is a quadratic equation. Since the coefficient of $x^2$ is negative, the parabola opens downwards, meaning it has a maximum point.

To find the number of unsold seats that maximize the revenue, we can use the formula for the vertex of a parabola:

$x = -\frac{b}{2a}$

where $a = -40$ and $b = 3{,}700$.

Substitute these values into the formula:

$x = -\frac{3{,}700}{2 \times -40} = \frac{3{,}700}{80} = 46.25$

Since the number of unsold seats must be an integer, we can round this to either $x = 46$ or $x = 47$.

Conclusion

• The number of unsold seats that will produce the maximum revenue is approximately 46 or 47.
• The total revenue expression is $R(x) = -40x^2 + 3{,}700x + 30{,}000$.

Would you like to see a calculation of the exact maximum revenue? Do you need further clarification?

Related Questions

1. How would the total revenue change if the fare per person without unsold seats increased by $20?
2. What is the total revenue if all 100 seats are sold?
3. How does the total revenue function change if the number of seats on the plane is increased to 120?
4. What happens to the maximum revenue if the additional charge per unsold seat is reduced to $30?
5. How would the optimal number of unsold seats change if the fare per person is reduced to $250?

Tip

To maximize revenue or minimize cost using quadratic functions, always look for the vertex, as it gives the optimum solution in parabolic models.