Suppose a computer manufacturer has the total cost function


C(x)=85x+9504 

and the total revenue function

 R(x)=382x  (in dollars).


(a) What is the equation of the profit function P(x) for this commodity?  (Simplify your answer)

P(x)=   

(b) What is the profit on 253 items?

 P(253)=  

(c) How many fans must be sold to avoid losing money?

 items

To solve the problem, let's address each part step by step:

### (a) Find the equation of the profit function $ P(x) $.

The profit function is calculated as:

\[
P(x) = R(x) - C(x)
\]

Given:
- The total cost function $ C(x) = 85x + 9504 $
- The total revenue function $ R(x) = 382x $

Now, substitute these into the profit function:

\[
P(x) = 382x - (85x + 9504)
\]

Simplify the equation:

\[
P(x) = 382x - 85x - 9504
\]

\[
P(x) = 297x - 9504
\]

So, the equation of the profit function is:

\[
P(x) = 297x - 9504
\]

### (b) Find the profit on 253 items.

To find the profit when $ x = 253 $, substitute $ x = 253 $ into the profit function:

\[
P(253) = 297(253) - 9504
\]

Now, calculate the value:

\[
P(253) = 75141 - 9504 = 65637
\]

So, the profit on 253 items is:

\[
P(253) = 65637 \text{ dollars}
\]

### (c) How many items must be sold to avoid losing money?

To avoid losing money, the profit must be at least 0, i.e., $ P(x) \geq 0 $. Set the profit function to 0 and solve for $ x $:

\[
297x - 9504 = 0
\]

Solve for $ x $:

\[
297x = 9504
\]

\[
x = \frac{9504}{297} = 32
\]

So, the manufacturer must sell at least 32 items to avoid losing money.

---

If you have any questions or need more details, feel free to ask! Here are some related questions:

1. How would the profit function change if the total cost per item increases to $100?
2. What is the break-even point for this scenario?
3. What is the revenue when selling 253 items?
4. How would the profit change if the fixed cost increases to $10,000?
5. How does the total cost function compare to a scenario with a different fixed cost?

**Tip:** To better understand profit calculations, always keep track of fixed costs and variable costs separately.

Suppose a computer manufacturer has the total cost function C(x) = 85x + 9504 and the total revenue function R(x) = 382x (in dollars). (a) What is the equation of the profit function P(x) for this commodity? (Simplify your answer) P(x) = ? (b) What is the profit on 253 items? P(253) = ? (c) How many fans must be sold to avoid losing money? items

Learn how to calculate the profit function for a computer manufacturer with the cost function C(x) = 85x + 9504 and revenue function R(x) = 382x. This step-by-step guide explains how to find the profit equation, calculate profit on 253 items, and determine the break-even point. Ideal for students in Grades 10-12.

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