To find the revenue equation, we need to establish a linear relationship using the given information. Here's how we approach this:

Step 1: Revenue Equation Formula

The revenue equation has the form: $R(x) = p \cdot x$ where:

• $R(x)$ is the total revenue.
• $p$ is the price per item.
• $x$ is the number of items sold.

Step 2: Profit Relationship

Profit ($P(x)$) is given by: $P(x) = R(x) - C(x)$ where:

• $C(x)$ is the total cost function.
• $R(x)$ is the total revenue.

Step 3: Cost Function

The total cost function $C(x)$ is: $C(x) = 15x + 3700$ Here:

• $15x$: Variable cost (cost per item multiplied by the number of items).
• $3700$: Fixed costs.

Step 4: Profit at 200 Items

When $x = 200$, the profit is $4300. Substituting into the profit equation: $P(200) = R(200) - C(200)$ Substitute $C(200)$ into the equation: $C(200) = 15(200) + 3700 = 3000 + 3700 = 6700$ Now substitute the known values: $4300 = R(200) - 6700$ $R(200) = 4300 + 6700 = 11000$

Step 5: Determine Price per Item

From the revenue equation $R(x) = p \cdot x$, when $x = 200$, $R(200) = 11000$. Solve for $p$: $p = \frac{R(200)}{200} = \frac{11000}{200} = 55$

Step 6: Final Revenue Equation

The revenue equation is: $R(x) = 55x$

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Let me know if you want additional details or clarification! Here are some related questions:

1. How can the break-even point be determined from the cost and revenue equations?
2. What would the profit function $P(x)$ look like for this scenario?
3. How does the profit change if the cost per item increases to $18?
4. What is the revenue when 300 items are sold?
5. How does the graph of $R(x)$ compare to the graph of $C(x)$?

Tip: Always verify relationships between cost, revenue, and profit when interpreting linear models.