Let's break down the problem step by step and address each part:

(a) Finding the linear revenue function $R(x)$

The revenue function $R(x)$ is a linear equation of the form: $R(x) = mx + b$ where:

• $x$ represents the sales volume (number of items sold),
• $m$ is the revenue per item (the slope),
• $b$ is the fixed revenue when $x = 0$ (the y-intercept).

We are given:

• At $x = 1465$, $R(1465) = 32,962.50$.

Since we only have one point for the revenue function, we assume that the y-intercept $b$ is 0, meaning that the revenue is entirely variable based on sales. Thus: $R(x) = mx$ Using the point $(1465, 32,962.50)$, we find the slope $m$: $m = \frac{R(1465)}{1465} = \frac{32,962.50}{1465} = 22.50$ So, the revenue function is: $R(x) = 22.50x$

(b) Finding the linear cost function $C(x)$

The cost function $C(x)$ is also a linear equation of the form: $C(x) = mx + b$ We are given two points for the cost function:

1. $C(1465) = 26,312.50$,
2. $C(920) = 17,538$.

We can use these two points to find the slope $m$ and the intercept $b$.

First, calculate the slope $m$: $m = \frac{C(1465) - C(920)}{1465 - 920} = \frac{26,312.50 - 17,538}{1465 - 920} = \frac{8,774.50}{545} = 16.10$ Now, use one of the points (e.g., $(920, 17,538)$) to solve for the intercept $b$: $C(920) = 16.10(920) + b = 17,538$ $17,538 = 14,812 + b$ $b = 17,538 - 14,812 = 2,726$ Thus, the cost function is: $C(x) = 16.10x + 2,726$

(c) Finding the break-even point

The break-even point occurs when the revenue equals the cost, i.e., when $R(x) = C(x)$. Set the two functions equal to each other: $22.50x = 16.10x + 2,726$ Solve for $x$: $22.50x - 16.10x = 2,726$ $6.40x = 2,726$ $x = \frac{2,726}{6.40} = 426.5625$

Since the number of items must be a whole number, round up to the nearest whole number: $x = 427 \text{ items}$

Thus, the break-even point is at 427 items.

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Would you like more details on any part of this solution? Here are 5 related questions that you might want to explore:

1. How would the break-even point change if the cost per item increased?
2. What happens to the break-even point if the fixed costs decrease?
3. How do changes in the selling price per item affect the revenue function?
4. How would you determine the profit function from the revenue and cost functions?
5. Can a company still be profitable even if they haven’t reached the break-even point?

Tip: Always ensure to clearly define what your variables represent when setting up equations from word problems to avoid confusion.