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.