To determine who has the correct solution, let's analyze the given revenue function and calculate the price that maximizes it.

The revenue function provided is: $R = -12p^2 + 600p + 15000$

This function is a quadratic equation in terms of $p$ (the price), where:

• The coefficient of $p^2$ is $a = -12$,
• The coefficient of $p$ is $b = 600$,
• The constant term is $c = 15000$.

Since the coefficient of $p^2$ is negative, the parabola opens downward, meaning there is a maximum point.

Finding the Vertex (Maximum Revenue Point)

The formula for the vertex of a quadratic equation $ax^2 + bx + c$ is given by: $p = -\frac{b}{2a}$

Substituting the values: $p = -\frac{600}{2 \times -12} = \frac{600}{24} = 25$

Conclusion

• Ben's solution is correct. He calculated $p = 25$, which is the price that maximizes the revenue.
• Arem's solution contains an error, as he calculated $p = 30$, which is incorrect.

Final Answer

The correct answer is: Ben

Would you like further details on the calculation or the reasoning?

Here are five related questions for further exploration:

1. How would the maximum revenue change if the revenue function had a different coefficient for $p^2$?
2. What if the revenue function opened upwards? How would that affect the revenue?
3. Can we determine the maximum revenue amount at $p = 25$?
4. How would we interpret the vertex in a cost function instead of a revenue function?
5. How would you solve this problem using calculus instead of algebra?

Tip: For quadratic functions, the vertex formula $-\frac{b}{2a}$ quickly identifies the maximum or minimum without needing to complete the square.