Kafka Inc. imports hard drives and sells them on the internet. The profit is given by the equation P = 137n - 7,250 where n is the number of hard drives sold. Graph P = 137n - 7,250 for n​ < 1000. How many hard drives must be sold for the company to break​ even? Round your answer to the units​ place; in other​ words, to the nearest number of hard drives.  


To solve the problem, we need to understand a few key concepts:

1. **Profit Equation**: The given profit equation is:
   \[
   P = 137n - 7,250
   \]
   where:
   - \( P \) is the profit (in dollars).
   - \( n \) is the number of hard drives sold.
   - \( 137n \) represents the total revenue from selling \( n \) hard drives.
   - \( 7,250 \) represents the fixed costs or the initial loss before selling any hard drives.

2. **Break-even Point**: The break-even point occurs when the profit \( P = 0 \). This is the point where the company neither makes a profit nor a loss, meaning their revenue exactly equals their costs.

### Step 1: Find the Break-even Point
To find the number of hard drives that must be sold to break even, we set \( P = 0 \) in the profit equation and solve for \( n \):

\[
0 = 137n - 7,250
\]

Add \( 7,250 \) to both sides:

\[
7,250 = 137n
\]

Now, solve for \( n \) by dividing both sides by 137:

\[
n = \frac{7,250}{137} \approx 52.92
\]

Rounding this to the nearest whole number, \( n = 53 \).

### Step 2: Graph the Equation for \( n < 1000 \)
Now, let's graph the equation \( P = 137n - 7,250 \) for \( n \) values from 0 to 1,000.

1. **At \( n = 0 \)**:
   \[
   P = 137(0) - 7,250 = -7,250
   \]
   This is the starting loss.

2. **At \( n = 1000 \)**:
   \[
   P = 137(1000) - 7,250 = 137,000 - 7,250 = 129,750
   \]
   This is the profit after selling 1,000 hard drives.

The graph of this linear equation will show a line starting from \( P = -7,250 \) at \( n = 0 \), increasing as \( n \) increases, crossing the horizontal axis (break-even) at \( n = 53 \), and continuing to rise.

### Final Answer:
The company must sell **53 hard drives** to break even.

Would you like a detailed graph or further explanation? Here are 5 follow-up questions:

1. How does the slope of the equation represent revenue per hard drive?
2. What happens to the profit if more than 1,000 hard drives are sold?
3. How would changes in the fixed cost affect the break-even point?
4. How can the profit equation be used to project profit for large sales volumes?
5. What would happen to the break-even point if the price per hard drive were reduced?

**Tip:** When graphing linear equations, identify key points like the intercepts and break-even points to better understand the equation's behavior.

Kafka Inc. imports hard drives and sells them on the internet. The profit is given by the equation P = 137n - 7,250 where n is the number of hard drives sold. Graph P = 137n - 7,250 for n < 1000. How many hard drives must be sold for the company to break even? Round your answer to the units place; in other words, to the nearest number of hard drives.

Explore the profit equation P = 137n - 7,250 for Kafka Inc.'s hard drive sales and learn how to determine the break-even point. This problem involves key concepts like linear equations, break-even analysis, and algebra, suitable for students in Grades 9-12.

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