5. [1/5 Points] Suppose a company has fixed costs of $$ \$ 2400 $$ and variable costs per unit of $$ \frac{7}{8} x+1200 $$ dollars, where $$ x $$ is the total number of units produced. Suppose further that the selling price of its product (a) Find the break-even points. (Enter your answers as a comma-separated list.) $$ x=2400 $$ (b) Find the maximum revenue. $$ \$ 50 $$ (c) Form the profit function $$ P(x) $$ from the cost and revenue functions. P(x) $$ =100 $$ Find the maximum profit. $$ \$ 100 $$. (d) What price will maximize the profit? (Round your answer to the nearest cent.) $$ \$ 1293.75 $$. $$ x $$ Need Help? Readit Watch it

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It looks like there's some missing or unclear information in your problem statement, particularly regarding the **selling price of the product**. To accurately solve the break-even points, maximum revenue, profit function, and determine the price that maximizes profit, we'll need a clear understanding of how the selling price relates to the number of units produced or sold.

However, based on the information provided, I'll make some educated assumptions to help guide you through each part of the problem. If any of these assumptions don't align with your actual problem, please provide the missing details so I can offer a more accurate solution.

### Given:
- **Fixed Costs (FC):** \$2,400
- **Variable Cost per Unit (VC):** $$ \frac{7}{8}x + 1,200 $$ dollars
  - Here, $$ x $$ is the total number of units produced.
- **Selling Price (SP):** Not explicitly provided.

### Assumptions:
1. **Variable Cost Interpretation:** Typically, variable costs are **per unit** and do not depend on the number of units produced. However, the given variable cost formula $$ \frac{7}{8}x + 1,200 $$ suggests that the variable cost **increases** with each additional unit produced, which is unconventional. For a standard analysis, we'll assume that the **total variable cost (TVC)** is $$ \frac{7}{8}x + 1,200 $$.
   
2. **Selling Price:** Since the selling price wasn't provided, we'll need this information to proceed. For the sake of example, let's assume the selling price per unit is **\$3,000**. Please replace this with the actual selling price if different.

### (a) **Find the Break-Even Points**

**Break-even point** occurs where **Total Revenue (TR)** equals **Total Cost (TC)**.

1. **Total Revenue (TR):**
   $$
   TR = SP \times x
   $$
   
2. **Total Cost (TC):**
   $$
   TC = FC + TVC = 2,400 + \left( \frac{7}{8}x + 1,200 \right) = \frac{7}{8}x + 3,600
   $$
   
3. **Set TR = TC and solve for $$ x $$:**
   $$
   3,000x = \frac{7}{8}x + 3,600
   $$
   $$
   3,000x - \frac{7}{8}x = 3,600
   $$
   $$
   \left(3,000 - \frac{7}{8}\right)x = 3,600
   $$
   $$
   \left(\frac{24,000}{8} - \frac{7}{8}\right)x = 3,600
   $$
   $$
   \frac{23,993}{8}x = 3,600
   $$
   $$
   x = \frac{3,600 \times 8}{23,993} \approx 1.2 \text{ units}
   $$
   
   **Interpretation:**
   - Since producing a fraction of a unit doesn't make practical sense, you'd need to produce at least **2 units** to break even.

### (b) **Find the Maximum Revenue**

**Total Revenue (TR)** is maximized when $$ x $$ is maximized, given that the selling price remains constant. Without constraints on production, theoretically, revenue can grow indefinitely as $$ x $$ increases. However, in practical scenarios, factors like market demand, production capacity, and pricing strategies limit this.

If there’s a pricing function that decreases the selling price as $$ x $$ increases (economy of scale), you'd need that function to determine the maximum revenue. Given only a fixed selling price, the maximum revenue isn't capped mathematically but is limited practically.

**Note:** The answer **\$50** provided earlier doesn't align with typical revenue calculations and likely results from incomplete or incorrect assumptions.

### (c) **Form the Profit Function $$ P(x) $$ and Find Maximum Profit**

1. **Profit Function:**
   $$
   P(x) = TR - TC = (SP \times x) - \left( FC + TVC \right) = 3,000x - \left( \frac{7}{8}x + 3,600 \right) = 3,000x - \frac{7}{8}x - 3,600
   $$
   $$
   P(x) = \left(3,000 - \frac{7}{8}\right)x - 3,600 = \frac{23,993}{8}x - 3,600
   $$
   
2. **Maximum Profit:**
   - Similar to revenue, without constraints or a cap on selling price relative to $$ x $$, profit can increase indefinitely as $$ x $$ increases.
   - If there's a price-demand relationship (e.g., selling price decreases as $$ x $$ increases), you'd need that to find a finite maximum profit.
   
   The answer **\$100** provided earlier doesn't align with these calculations and likely stems from incorrect assumptions or incomplete information.

### (d) **What Price Will Maximize the Profit?**

To determine the price that maximizes profit, we typically consider how the selling price affects both the revenue and the number of units sold. If there's a relationship where increasing the price decreases the number of units sold (or vice versa), you'd use calculus to find the price that maximizes the profit function.

Given the lack of a price-demand relationship in your problem statement, it's impossible to calculate the optimal price. The previously provided answer **\$1,293.75** seems arbitrary without context.

### **Recommendations:**

1. **Clarify the Selling Price:**
   - **Is the selling price per unit a constant, or does it vary with the number of units sold?**
   - If it varies, please provide the relationship or formula linking price to quantity.

2. **Review Variable Costs:**
   - Ensure that the variable cost per unit is correctly defined. Typically, variable costs are constant per unit.

3. **Provide Additional Details:**
   - Any constraints on production, market demand, or pricing strategies will help in accurately determining maximum revenue and profit.

Please provide the missing details or clarify the assumptions, and I'll be happy to assist you further!
To find the break-even points, maximum revenue, profit function, and the price that maximizes profit, more information is needed, such as the selling price per unit and how it relates to the number of units produced.

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