Let C(x) be the cost to produce x batches of widgets, and let R(x) be the revenue in thousands of dollars. R(x)=-x^2+8x, C(x)=x+6. Find the minimum​ break-even quantity.
Using the expressions -x^2+8x and/or x+6, identify an equation to be solved in order to find the minimum​ break-even quantity. Find the maximum revenue.
How can the maximum revenue be​ found? The maximum revenue is...

### Break-even Quantity:

To find the **minimum break-even quantity**, we need to set the cost equal to the revenue:
\[
R(x) = C(x)
\]
Given:
\[
R(x) = -x^2 + 8x \quad \text{and} \quad C(x) = x + 6
\]
Equating the two:
\[
-x^2 + 8x = x + 6
\]
This simplifies to:
\[
-x^2 + 8x - x - 6 = 0
\]
\[
-x^2 + 7x - 6 = 0
\]
Multiplying through by -1 to simplify:
\[
x^2 - 7x + 6 = 0
\]
Now, solve this quadratic equation:
\[
(x - 6)(x - 1) = 0
\]
Thus, the solutions are:
\[
x = 6 \quad \text{or} \quad x = 1
\]
The minimum break-even quantity is the smaller solution:
\[
\boxed{x = 1}
\]

### Maximum Revenue:

To find the maximum revenue, we need to maximize the revenue function \(R(x) = -x^2 + 8x\), which is a quadratic function. Since the coefficient of \(x^2\) is negative, the parabola opens downwards, and the maximum occurs at the vertex.

The formula for the x-coordinate of the vertex is:
\[
x = -\frac{b}{2a}
\]
For \(R(x) = -x^2 + 8x\), \(a = -1\) and \(b = 8\). Plugging these into the formula:
\[
x = -\frac{8}{2(-1)} = 4
\]
Thus, the maximum revenue occurs when \(x = 4\).

To find the maximum revenue, substitute \(x = 4\) into \(R(x)\):
\[
R(4) = -(4)^2 + 8(4) = -16 + 32 = 16
\]
Since revenue is measured in thousands of dollars, the maximum revenue is:
\[
\boxed{16 \text{ thousand dollars}}
\]

### Summary:

- **Minimum break-even quantity**: \(x = 1\)
- **Maximum revenue**: \(16\) thousand dollars

### Steps for Maximum Revenue:

1. Identify the revenue function \(R(x) = -x^2 + 8x\).
2. Use the vertex formula \(x = -\frac{b}{2a}\) to find the value of \(x\) that maximizes revenue.
3. Substitute \(x = 4\) into the revenue function to calculate the maximum revenue.

---

Would you like further details or have any questions about the process?

Here are five follow-up questions you could explore:
1. What is the break-even quantity when the maximum revenue is reached?
2. How does the profit relate to the break-even quantities?
3. What would happen if the cost function changed to \(C(x) = x + 8\)?
4. Can you graph both the revenue and cost functions to visualize the break-even points?
5. How does the quadratic nature of the revenue function impact business decisions?

**Tip**: Always check if the quadratic function opens upwards or downwards when determining maxima or minima!

Let C(x) be the cost to produce x batches of widgets, and let R(x) be the revenue in thousands of dollars. R(x)=-x^2+8x, C(x)=x+6. Find the minimum​ break-even quantity. Using the expressions -x^2+8x and/or x+6, identify an equation to be solved in order to find the minimum​ break-even quantity. Find the maximum revenue. How can the maximum revenue be​ found? The maximum revenue is...

Learn how to calculate the minimum break-even quantity and maximum revenue for widget production. This problem covers essential algebraic concepts, including solving quadratic equations and analyzing cost-revenue relationships. Ideal for students in Grades 9-12.

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