https://file.aiquickdraw.com/mathbot/file/17262143846193zz22mft.jpg

The problem in the image asks:

"A bus company will charter a bus that holds 50 people to groups of 35 or more. If a group contains exactly 35 people, each person pays $60. In larger groups, everybody's fare is reduced by $1 for each person in excess of 35. Determine the size of the group for which the bus company's revenue will be greatest."

Let's break down the situation and create an equation for the bus company's revenue:

### Variables
- Let $ x $ be the number of people in the group above 35. So the total number of people in the group is $ 35 + x $.
- The fare is reduced by $1 for each person above 35. If there are $ x $ extra people, the fare per person is $ 60 - x $.

### Revenue Equation
The total revenue $ R $ is the number of people in the group multiplied by the fare they each pay:
\[
R = (35 + x)(60 - x)
\]

### Expand the equation:
\[
R = 35(60 - x) + x(60 - x)
\]
\[
R = 2100 - 35x + 60x - x^2
\]
\[
R = 2100 + 25x - x^2
\]

### Maximizing the Revenue
To find the value of $ x $ that maximizes the revenue, we take the derivative of $ R $ with respect to $ x $ and set it equal to zero:
\[
\frac{dR}{dx} = 25 - 2x
\]
Set the derivative equal to zero:
\[
25 - 2x = 0
\]
Solve for $ x $:
\[
x = 12.5
\]

Since $ x = 12.5 $ is not an integer, we evaluate the revenue at both $ x = 12 $ and $ x = 13 $, because the group size must be a whole number.

For $ x = 12 $:
\[
R = (35 + 12)(60 - 12) = 47 \times 48 = 2256
\]

For $ x = 13 $:
\[
R = (35 + 13)(60 - 13) = 48 \times 47 = 2256
\]

Both values give the same revenue, so the size of the group that maximizes the company's revenue is either **47 or 48 people**.

---

Would you like further details or clarification on any of the steps? Here are some follow-up questions you might find interesting:

1. Why is it necessary to check both $ x = 12 $ and $ x = 13 $ after solving for $ x = 12.5 $?
2. How would the revenue equation change if the fare reduction per person was more than $1?
3. What happens to the revenue if the group exceeds the bus capacity of 50 people?
4. Could the problem be solved using a different method, such as completing the square?
5. How would the results change if the base fare per person for 35 people was not $60?

**Tip:** Always check your boundary values when working with optimization problems involving real-world constraints!

A bus company will charter a bus that holds 50 people to groups of 35 or more. If a group contains exactly 35 people, each person pays $60. In large groups, everybody's fare is reduced by $1 for each person in excess of 35. Determine the size of the group for which the bus company's revenue will be greatest.

Solve a real-world revenue maximization problem where a bus company reduces the fare by $1 for each additional passenger beyond 35. Learn how to use quadratic equations and derivatives to determine the optimal group size that maximizes revenue. Suitable for Grades 9-12.

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