To determine the break-even point for a business given the revenue function $R(x)$ and the cost function $C(x)$, we need to find the value of $x$ such that $R(x) = C(x)$.

The given functions are: $R(x) = 200x - x^2$ $C(x) = 25x + 7150$

Setting $R(x)$ equal to $C(x)$ gives the equation: $200x - x^2 = 25x + 7150$

Rearranging and simplifying this equation: $x^2 - 175x + 7150 = 0$

We can solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = -175$, and $c = 7150$.

Let's calculate the values of $x$.The solutions for $x$ are 65 and 110. However, we are given that the functions are valid only for $0 \leq x \leq 100$. Thus, the valid break-even point within this range is when the manufacturer produces 65 units.

Would you like more details or have any further questions on this topic?

Here are five related questions you might find interesting:

1. How would the break-even point change if the selling price per unit increased?
2. What is the profit when producing at the break-even point?
3. How does a change in the fixed cost $(7150)$ affect the break-even point?
4. Can you explain how to derive the revenue and cost functions from given data?
5. What is the significance of the discriminant in determining the nature of the roots for the quadratic equation in this context?

Tip: When checking the viability of solutions in a real-world scenario, always consider the constraints given, such as the range of $x$ values in this case.