To find the number of units needed for the manufacturer to break even, we need to determine when the total revenue equals the total cost.

We are given:

• Total Revenue: $R(x) = 126.50x$
• Total Cost: $C(x) = 44.90x + 9710.40$

At the break-even point: $R(x) = C(x)$ Substitute the expressions for $R(x)$ and $C(x)$: $126.50x = 44.90x + 9710.40$

Now, solve for $x$:

1. Subtract $44.90x$ from both sides: $126.50x - 44.90x = 9710.40$ $81.60x = 9710.40$

2. Divide both sides by 81.60 to isolate $x$: $x = \frac{9710.40}{81.60} \approx 119$

Thus, the manufacturer must produce and sell 119 units to break even.

Would you like more details or have any questions?

Here are 5 related questions you might find helpful:

1. How does changing the fixed cost affect the break-even point?
2. What if the revenue per unit is increased by $10? How does that impact the break-even number?
3. How does a variable cost reduction affect the total cost function?
4. What is the significance of the slope of the total revenue function?
5. Can the break-even point be negative in any scenario?

Tip: Always analyze both fixed and variable costs when calculating the break-even point to understand the full financial picture.