Sure, let's break down this problem step-by-step.
### Problem Statement:
Recycled CDs, Incorporated, offers a deal where you get 5 used CDs for [tex]$22, and each additional CD (above the initial 5) costs $[/tex]5. We need to create a cost function, [tex]\( C(x) \)[/tex], that represents the total cost of purchasing [tex]\( x \)[/tex] additional CDs beyond the initial 5. Then, we need to find the cost of buying 12 CDs in total.
### Step-by-Step Solution:
1. Identify the given values:
- Base number of CDs: 5 CDs (which comes with a fixed cost).
- Base cost for these 5 CDs: [tex]$22.
- Cost for each additional CD: $[/tex]5.
- Total number of CDs desired: 12 CDs.
2. Define the variable:
- Let [tex]\( x \)[/tex] represent the number of CDs beyond the initial 5.
- So, [tex]\( x = \text{total number of CDs} - 5 \)[/tex].
3. Calculate the number of additional CDs:
- Total CDs desired = 12
- Number of additional CDs, [tex]\( x \)[/tex] = 12 - 5 = 7
4. Develop the cost function [tex]\( C(x) \)[/tex]:
- For the first 5 CDs, the cost is fixed at [tex]$22.
- For each additional CD, the cost is $[/tex]5 per CD.
- Therefore, the total cost [tex]\( C(x) \)[/tex], where [tex]\( x \)[/tex] represents the number of additional CDs, is given by:
[tex]\[
C(x) = 22 + 5x
\][/tex]
5. Find the cost of buying 12 CDs:
- We already calculated [tex]\( x = 7 \)[/tex].
- Substitute [tex]\( x = 7 \)[/tex] into the cost function [tex]\( C(x) \)[/tex]:
[tex]\[
C(7) = 22 + 5 \times 7
\][/tex]
- Perform the arithmetic:
[tex]\[
C(7) = 22 + 35 = 57
\][/tex]
### Final Answer:
- The cost function for purchasing [tex]\( x \)[/tex] additional CDs is:
[tex]\[
C(x) = 22 + 5x
\][/tex]
- The cost of buying 12 CDs will be [tex]$57.
So, the detailed step-by-step solution leads us to the conclusion that the cost of purchasing 12 CDs is $[/tex]57.
