Please Answer these questions:

1. A company sells its product for $46 per unit. Write an expression for the amount of money received (revenue *R*) from the sale of *x* units of the product.

*R* = 



2. Suppose a calculator manufacturer has the total cost function 

*C*(*x*) = 20*x* + 6200

 and the total revenue function 

*R*(*x*) = 66*x*.

(a) What is the equation of the profit function *P*(*x*) for the calculator?
*P*(*x*) = 
 


(b) What is the profit on 2400 units?
*P*(2400) = $ 


3. Suppose a ceiling fan manufacturer has the total cost function 

*C*(*x*) = 35*x* + 1320

 and the total revenue function 

*R*(*x*) = 68*x*.

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


*P*(*x*) = 
 


(b) What is the profit on 20 units?


*P*(20) = 



Interpret your result.


The total costs are less than the revenue.The total costs are more than the revenue.    The total costs are exactly the same as the revenue.


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

4. Suppose a computer manufacturer has the total cost function 

*C*(*x*) = 72*x* + 3600

 (in dollars) and the total revenue function 

*R*(*x*) = 372*x*

 (in dollars).

(a) What is the equation of the profit function *P*(*x*) (in dollars) for this commodity?
*P*(*x*) = 



(b) What is the profit (in dollars) on 348 items?
*P*(348) = $ 

(c) How many items must be sold to avoid losing money?
*x* =  items

### 1. Revenue Expression

The revenue (*R*) is calculated as the price per unit multiplied by the number of units sold. If the price is $46 per unit and *x* represents the number of units sold, the revenue function is:

\[
R(x) = 46x
\]

### 2. Calculator Manufacturer

The total cost function is $ C(x) = 20x + 6200 $, and the total revenue function is $ R(x) = 66x $.

#### (a) Profit Function
The profit function (*P(x)*) is the difference between the total revenue (*R(x)*) and the total cost (*C(x)*):
\[
P(x) = R(x) - C(x)
\]
Substitute the given functions for *R(x)* and *C(x)*:
\[
P(x) = 66x - (20x + 6200)
\]
Simplify:
\[
P(x) = 46x - 6200
\]

#### (b) Profit on 2400 Units
Substitute *x = 2400* into the profit function:
\[
P(2400) = 46(2400) - 6200
\]
\[
P(2400) = 110400 - 6200
\]
\[
P(2400) = 104200
\]
So, the profit on 2400 units is:
\[
P(2400) = \$104,200
\]

### 3. Ceiling Fan Manufacturer

The total cost function is $ C(x) = 35x + 1320 $, and the total revenue function is $ R(x) = 68x $.

#### (a) Profit Function
The profit function (*P(x)*) is given by:
\[
P(x) = R(x) - C(x)
\]
Substitute the given functions for *R(x)* and *C(x)*:
\[
P(x) = 68x - (35x + 1320)
\]
Simplify:
\[
P(x) = 33x - 1320
\]

#### (b) Profit on 20 Units
Substitute *x = 20* into the profit function:
\[
P(20) = 33(20) - 1320
\]
\[
P(20) = 660 - 1320
\]
\[
P(20) = -660
\]
The profit on 20 units is $-660$, meaning the company loses \$660 on 20 units.

#### Interpretation:
Since the profit is negative, the total costs are more than the revenue.

#### (c) Number of Fans to Avoid Losing Money
To avoid losing money, the profit should be zero or greater:
\[
P(x) = 33x - 1320 \geq 0
\]
Solve for *x*:
\[
33x \geq 1320
\]
\[
x \geq \frac{1320}{33}
\]
\[
x \geq 40
\]
So, the company must sell at least 40 fans to avoid losing money.

### 4. Computer Manufacturer

The total cost function is $ C(x) = 72x + 3600 $, and the total revenue function is $ R(x) = 372x $.

#### (a) Profit Function
The profit function (*P(x)*) is:
\[
P(x) = R(x) - C(x)
\]
Substitute the given functions for *R(x)* and *C(x)*:
\[
P(x) = 372x - (72x + 3600)
\]
Simplify:
\[
P(x) = 300x - 3600
\]

#### (b) Profit on 348 Items
Substitute *x = 348* into the profit function:
\[
P(348) = 300(348) - 3600
\]
\[
P(348) = 104400 - 3600
\]
\[
P(348) = 100800
\]
The profit on 348 items is:
\[
P(348) = \$100,800
\]

#### (c) Number of Items to Avoid Losing Money
To avoid losing money, the profit should be zero or greater:
\[
P(x) = 300x - 3600 \geq 0
\]
Solve for *x*:
\[
300x \geq 3600
\]
\[
x \geq \frac{3600}{300}
\]
\[
x \geq 12
\]
So, the company must sell at least 12 items to avoid losing money.

---

### Follow-up Questions:
1. What happens to the profit function when fixed costs increase?
2. How would the ceiling fan manufacturer's profit function change if the cost per fan increases to $40?
3. What is the break-even point if the calculator's price drops to $50?
4. How would selling more than 40 ceiling fans affect profit?
5. How does a lower unit cost impact the computer manufacturer’s profitability?

#### Tip: Always analyze the break-even point carefully, as it determines the minimum sales needed for a business to cover its costs.

Please Answer these questions:

1. A company sells its product for $46 per unit. Write an expression for the amount of money received (revenue R) from the sale of x units of the product.

2. Suppose a calculator manufacturer has the total cost function C(x) = 20x + 6200 and the total revenue function R(x) = 66x.

(a) What is the equation of the profit function P(x) for the calculator?
(b) What is the profit on 2400 units?

3. Suppose a ceiling fan manufacturer has the total cost function C(x) = 35x + 1320 and the total revenue function R(x) = 68x.

(a) What is the equation of the profit function P(x) for this commodity?
(b) What is the profit on 20 units?
Interpret your result.
(c) How many fans must be sold to avoid losing money?

4. Suppose a computer manufacturer has the total cost function C(x) = 72x + 3600 and the total revenue function R(x) = 372x.

(a) What is the equation of the profit function P(x)?
(b) What is the profit on 348 items?
(c) How many items must be sold to avoid losing money?

Explore how to solve business-related algebra problems involving revenue, cost, and profit functions. This lesson includes step-by-step solutions for calculating profit based on given cost and revenue functions, suitable for students in Grades 9-11.

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