The profit of a​ company, in​ dollars, is the difference between the​ company's revenue and cost. The​ cost, C(x), and​ revenue, R(x), are functions for a particular company. The x represents the number of items produced and sold to distributors.

C(x)=2400+40x     R(x)=780x−x2                                                                                                                                                    Determine the number of items that must be produced and sold to obtain the maximum profit. The number of items that must be produced and sold to obtain the maximum profit is




To find the number of items $ x $ that must be produced and sold to obtain the maximum profit, we start by determining the profit function $ P(x) $. The profit $ P(x) $ is calculated as the difference between the revenue $ R(x) $ and the cost $ C(x) $:

\[
P(x) = R(x) - C(x)
\]

Given:
\[
C(x) = 2400 + 40x \quad \text{and} \quad R(x) = 780x - x^2
\]

Substitute these into the profit function:

\[
P(x) = (780x - x^2) - (2400 + 40x)
\]

Simplify:

\[
P(x) = 780x - x^2 - 2400 - 40x
\]
\[
P(x) = -x^2 + 740x - 2400
\]

This is a quadratic function in the form $ P(x) = ax^2 + bx + c $, where $ a = -1 $, $ b = 740 $, and $ c = -2400 $. Since the leading coefficient $ a $ is negative, the parabola opens downward, and the vertex represents the maximum point.

To find the vertex $ x $ of a parabola given by $ ax^2 + bx + c $, use the formula:

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

Substitute $ a = -1 $ and $ b = 740 $:

\[
x = -\frac{740}{2(-1)} = \frac{740}{2} = 370
\]

Thus, the number of items that must be produced and sold to obtain the maximum profit is **370**.

Would you like more details or have any other questions?

### Related Questions:
1. What is the maximum profit for producing and selling 370 items?
2. How does the revenue change if 500 items are produced instead?
3. What is the break-even point where the cost equals the revenue?
4. How many items should be produced to ensure a profit above $5,000?
5. How would the profit function change if the fixed cost increases to $3,000?

### Tip:
For quadratic functions $ ax^2 + bx + c $, if $ a < 0 $, the vertex will give you the maximum value of the function, while $ a > 0 $ will give the minimum.

The profit of a company, in dollars, is the difference between the company's revenue and cost. The cost, C(x), and revenue, R(x), are functions for a particular company. The x represents the number of items produced and sold to distributors. C(x) = 2400 + 40x and R(x) = 780x − x^2. Determine the number of items that must be produced and sold to obtain the maximum profit.

Explore how to determine the number of items that must be produced and sold to maximize profit using quadratic functions. Learn to apply the profit function P(x) = R(x) - C(x) and the vertex formula x = -b / 2a to solve this real-world algebraic problem. Ideal for students in Grades 9-12.

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