A manufacturer of automobile air conditioners has total revenue given by Upper R equals 126.50 xR=126.50x and total cost given by Upper C equals 44.90 x plus 9710.40C=44.90x+9710.40​, where x is the number of units produced and sold. Use a nongraphical method to find the number of units needed for the manufacturer to break even.
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Part 1
The manufacturer must produce and sell enter your response here units to break even.

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.

A manufacturer of automobile air conditioners has total revenue given by R = 126.50x and total cost given by C = 44.90x + 9710.40, where x is the number of units produced and sold. Use a nongraphical method to find the number of units needed for the manufacturer to break even.

Learn how to solve a break-even analysis problem for an automobile air conditioner manufacturer using linear equations. This problem involves finding when total revenue equals total cost, covering key algebraic concepts. Suitable for students in Grades 10-12.

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