A company finds that it can make a profit of P dollars each month by selling x patterns, according to the formula
P (x) = - 002x + 3.5x - 1300..
How many patterns must it sell each month to have a maximum profit?

The profit function is given as:

\[
P(x) = -0.002x^2 + 3.5x - 1300
\]

This is a quadratic function of the form:

\[
P(x) = ax^2 + bx + c
\]

Where:
- \(a = -0.002\)
- \(b = 3.5\)
- \(c = -1300\)

Since \(a < 0\), the parabola opens downwards, meaning there is a maximum point at the vertex of the parabola.

The formula for the vertex \(x\) of a quadratic function \(P(x) = ax^2 + bx + c\) is:

\[
x = \frac{-b}{2a}
\]

Substitute the values for \(a\) and \(b\):

\[
x = \frac{-3.5}{2(-0.002)} = \frac{-3.5}{-0.004} = 875
\]

Therefore, the company must sell **875 patterns** each month to achieve the maximum profit.

Would you like more details on this solution? 

Here are five related questions:
1. How can you calculate the maximum profit using this value of \(x\)?
2. What if the quadratic had a positive leading coefficient? How would it affect the result?
3. How do you find the vertex of a parabola geometrically?
4. How do we interpret the value of \(b\) in real-world problems?
5. What would happen to the profit function if fixed costs increased?

**Tip:** Always verify if a quadratic function opens upwards or downwards by checking the sign of the leading coefficient \(a\).

A company finds that it can make a profit of P dollars each month by selling x patterns, according to the formula P(x) = -0.002x^2 + 3.5x - 1300. How many patterns must it sell each month to have a maximum profit?

Discover how to determine the maximum profit for a company using the profit formula P(x) = -0.002x^2 + 3.5x - 1300. This solution guides you through the process of finding the vertex of a quadratic function, helping you identify the optimal number of patterns to sell. Ideal for students in Grades 9-11.

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