Question

A company produces very unusual CD's for which the variable cost is $ 8 per CD and the fixed costs are $ 50000. They will sell the CD's for $ 41 each. Let
be the number of CD's produced.
Write the total cost
as a function of the number of CD's produced.
C=$


Write the total revenue
as a function of the number of CD's produced.
R=$


Write the total profit
as a function of the number of CD's produced.
P=$


Find the number of CD's which must be produced to break even.
The number of CD's which must be produced to break even is

Answer 1

To solve this problem, we need to express the total cost, total revenue, and total profit as functions of the number of CDs produced, and then find the break-even point.

1. **Total Cost (C)**:
The total cost consists of fixed costs and variable costs. The total cost function can be formulated as:

\[
C = \text{Fixed Costs} + \text{Variable Cost per CD} \times \text{Number of CDs}
\]

Given that fixed costs are $50,000 and the variable cost per CD is $8, we can write:

\[
C = 50000 + 8x
\]

where \( x \) is the number of CDs produced.

2. **Total Revenue (R)**:
The total revenue is derived from selling the CDs. The total revenue function can be expressed as:

\[
R = \text{Price per CD} \times \text{Number of CDs}
\]

Given that the selling price is $41 per CD, we have:

\[
R = 41x
\]

3. **Total Profit (P)**:
The total profit is the difference between total revenue and total cost. The profit function can be written as:

\[
P = R - C
\]

Substituting in our revenue and cost equations, we get:

\[
P = 41x - (50000 + 8x)
\]
\[
P = 41x - 50000 - 8x
\]
\[
P = 33x - 50000
\]

4. **Break-Even Point**:
The break-even point occurs when total revenue equals total cost:

\[
R = C
\]

Plugging in our equations for revenue and cost:

\[
41x = 50000 + 8x
\]

Now, we solve for \( x \):

\[
41x - 8x = 50000
\]
\[
33x = 50000
\]
\[
x = \frac{50000}{33} \approx 1515.15
\]

Since the number of CDs produced must be a whole number, the company must produce at least 1516 CDs to break even.

### Summary:
1. Total cost function: \( C = 50000 + 8x \)
2. Total revenue function: \( R = 41x \)
3. Total profit function: \( P = 33x - 50000 \)
4. Break-even production: At least 1516 CDs.