Let's solve the given problem step-by-step:
### Part (a)
To find the average cost per disc, we need to include both the cost of the discs and the shipping charge.
1. Cost of discs: If each 100-pack costs \[tex]$33 and you purchase \( n \) packs, the total cost for the discs would be \( 33 \cdot n \) dollars.
2. Shipping charge: The shipping cost is a flat fee of \$[/tex]5.99 regardless of the number of packs ordered.
So, the total cost for [tex]\( n \)[/tex] 100-packs is:
[tex]\[ \text{Total cost} = 33n + 5.99 \][/tex]
Given that each 100-pack contains 100 discs, [tex]\( n \)[/tex] packs will contain [tex]\( 100n \)[/tex] discs. Thus, the average cost per disc is:
[tex]\[ C(n) = \frac{\text{Total cost}}{\text{Number of discs}} = \frac{33n + 5.99}{100n} \][/tex]
Therefore, the function [tex]\( C(n) \)[/tex] that represents the average cost per disc for your order is:
[tex]\[ C(n) = \frac{33n + 5.99}{100n} \][/tex]
### Part (b)
Next, we need to determine the minimum number of 100-packs that must be purchased for the average cost per disc to be equal to \[tex]$0.3317.
We set up the equation by equating the cost per disc function \( C(n) \) to 0.3317:
\[ \frac{33n + 5.99}{100n} = 0.3317 \]
To solve for \( n \), we clear the fraction by multiplying both sides by \( 100n \):
\[ 33n + 5.99 = 0.3317 \cdot 100n \]
\[ 33n + 5.99 = 33.17n \]
Next, isolate \( n \) by subtracting \( 33n \) from both sides:
\[ 5.99 = 33.17n - 33n \]
\[ 5.99 = 0.17n \]
Finally, solve for \( n \):
\[ n = \frac{5.99}{0.17} \approx 35.2352941176471 \]
### Part (c)
We need to determine the minimum number of 100 packs such that the average cost per disc is less than \$[/tex]0.3317. From part (b), we found that [tex]\( n = 35.2352941176471 \)[/tex] packs is needed for the average cost to be exactly \[tex]$0.3317.
Since \( n \) must be an integer and we want the average cost to be less than \$[/tex]0.3317, we round [tex]\( n \)[/tex] up to the next whole number:
[tex]\[ \text{Minimum number of 100 packs} = 36 \][/tex]
Thus, the minimum number of 100 packs that must be purchased for the average cost per disc to be less than \$0.3317 is [tex]\( \boxed{36} \)[/tex].
