A company produces very unusual CD's for which the variable cost is $$ \$ 8 $$ per CD and the fixed costs are $$ \$ $$ 45000. They will sell the CD's for $$ \$ 40 $$ each. Let $$ x $$ be the number of CD's produced. a. Write the total cost $$ C $$ as a function of the number of CD's produced. $$ C=\$ $$ b. Write the total revenue $$ R $$ as a function of the number of CD's produced. $$ R=\$ $$ c. Write the total profit $$ P $$ as a function of the number of CD's produced. $$ P=\$ $$ d. 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 Round to the nearest number of CDs.

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Let's solve each part step by step.

### a. Total Cost $$ C $$ as a Function of $$ x $$

The total cost $$ C(x) $$ consists of two parts:
- **Fixed Costs:** These are costs that do not change with the number of CDs produced. Here, the fixed cost is \$45,000.
- **Variable Costs:** These depend on the number of CDs produced. The variable cost is \$8 per CD.

So, the total cost function is:
$$
C(x) = 8x + 45,\!000
$$

### b. Total Revenue $$ R $$ as a Function of $$ x $$

Total revenue $$ R(x) $$ is calculated by multiplying the selling price per CD by the number of CDs sold. The selling price is \$40 per CD.

$$
R(x) = 40x
$$

### c. Total Profit $$ P $$ as a Function of $$ x $$

Profit $$ P(x) $$ is the difference between total revenue and total cost:
$$
P(x) = R(x) - C(x) = 40x - (8x + 45,\!000) = 32x - 45,\!000
$$

### d. Number of CDs to Break Even

To find the break-even point, set total revenue equal to total cost:
$$
40x = 8x + 45,\!000
$$
$$
40x - 8x = 45,\!000
$$
$$
32x = 45,\!000
$$
$$
x = \frac{45,\!000}{32} = 1,\!406.25
$$

Since you can't produce a fraction of a CD, round to the nearest whole number:
$$
x = 1,\!406 \text{ CDs}
$$

**Answer Summary:**
- **a.** $$ C(x) = 8x + 45,\!000 $$
- **b.** $$ R(x) = 40x $$
- **c.** $$ P(x) = 32x - 45,\!000 $$
- **d.** The company must produce **1,406 CDs** to break even.
a. Total cost $$ C(x) = 8x + 45,\!000 $$ b. Total revenue $$ R(x) = 40x $$ c. Total profit $$ P(x) = 32x - 45,\!000 $$ d. The company must produce **1,406 CDs** to break even.

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