 certain website has recently decided to start selling music albums online through a particular service. Users pay a fee to download an entire music album. Composer royalties and copyright fees cost an average of $4.50 per album, and the cost of operating and maintaining the service amounts to $600 per week. The company is currently charging customers $8.50 per album. (Let x denote the number of albums sold weekly and C(x), R(x), and P(x) denote the cost, revenue, and profit in dollars respectively for selling x albums in a week.)
(a)
What is the associated weekly cost function?
C(x) = 
 
What is the associated weekly revenue function?
R(x) = 
 
What is the associated weekly profit function?
P(x) = 
 
(b)
How many albums must be sold per week to make a profit?
more than 
 albums
(c)
If the charge is lowered to $7.00 per album, how many albums must be sold per week to make a profit?
more than 
 albums

Let's break down this problem step by step.

### Part (a) - Cost, Revenue, and Profit Functions

#### 1. **Weekly Cost Function (C(x))**

The cost function $ C(x) $ includes two components:
- The fixed cost of operating the service is \$600 per week.
- The variable cost, which is the composer royalties and copyright fees of \$4.50 per album.

So, the total weekly cost is:

\[
C(x) = 4.50x + 600
\]

where $ x $ is the number of albums sold per week.

#### 2. **Weekly Revenue Function (R(x))**

The revenue function $ R(x) $ is based on how much the company charges per album. Currently, they charge \$8.50 per album.

So, the total weekly revenue is:

\[
R(x) = 8.50x
\]

#### 3. **Weekly Profit Function (P(x))**

Profit is the difference between revenue and cost:

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

Substituting the expressions for $ R(x) $ and $ C(x) $:

\[
P(x) = 8.50x - (4.50x + 600)
\]

Simplifying this expression:

\[
P(x) = 8.50x - 4.50x - 600 = 4x - 600
\]

Thus, the profit function is:

\[
P(x) = 4x - 600
\]

### Part (b) - Number of Albums to Make a Profit

To make a profit, the profit function $ P(x) $ must be greater than 0:

\[
P(x) > 0
\]

Substituting the profit function:

\[
4x - 600 > 0
\]

Solving for $ x $:

\[
4x > 600
\]
\[
x > 150
\]

So, the company must sell **more than 150 albums per week** to make a profit.

### Part (c) - Lowering the Price to $7.00 Per Album

If the price is lowered to \$7.00 per album, the revenue function changes. The new revenue function becomes:

\[
R(x) = 7.00x
\]

The new profit function is:

\[
P(x) = 7.00x - (4.50x + 600)
\]

Simplifying:

\[
P(x) = 7.00x - 4.50x - 600 = 2.50x - 600
\]

To make a profit, $ P(x) > 0 $:

\[
2.50x - 600 > 0
\]

Solving for $ x $:

\[
2.50x > 600
\]
\[
x > 240
\]

Thus, the company must sell **more than 240 albums per week** to make a profit if the price is lowered to \$7.00 per album.

### Summary:

- **(a)**
  - Cost function: $ C(x) = 4.50x + 600 $
  - Revenue function: $ R(x) = 8.50x $
  - Profit function: $ P(x) = 4x - 600 $
- **(b)** To make a profit, more than 150 albums must be sold per week.
- **(c)** If the price is lowered to \$7.00, more than 240 albums must be sold per week to make a profit.

---

Would you like more details or clarification on any part of this solution? Here are some related questions to consider:

1. How would a change in the fixed cost affect the profit threshold?
2. What happens if composer royalties increase by 10%?
3. How many albums would need to be sold if the price was increased to \$9.00?
4. Can the company break even at a lower price if the fixed cost is reduced?
5. How would adding an additional cost of \$1 per album for marketing change the functions?

**Tip:** Always check the break-even point before lowering prices, as it can significantly affect profitability.

A certain website has recently decided to start selling music albums online through a particular service. Users pay a fee to download an entire music album. Composer royalties and copyright fees cost an average of $4.50 per album, and the cost of operating and maintaining the service amounts to $600 per week. The company is currently charging customers $8.50 per album. (Let x denote the number of albums sold weekly and C(x), R(x), and P(x) denote the cost, revenue, and profit in dollars respectively for selling x albums in a week.)
(a) What is the associated weekly cost function? C(x) =
What is the associated weekly revenue function? R(x) =
What is the associated weekly profit function? P(x) =
(b) How many albums must be sold per week to make a profit?
more than albums
(c) If the charge is lowered to $7.00 per album, how many albums must be sold per week to make a profit? more than albums

Explore the cost, revenue, and profit functions for a website selling music albums online. Learn how to calculate the weekly cost (C(x) = 4.50x + 600), revenue (R(x) = 8.50x), and profit (P(x) = 4x - 600) functions. Discover how many albums must be sold per week to turn a profit. Suitable for Grades 9-12.

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