To find the cost function for Jenna's clothing store per month, we need to account for both the variable costs (the costs that depend on the number of [tex]$T$[/tex]-shirts sold) and the fixed costs (the costs that do not depend on the number of [tex]$T$[/tex]-shirts sold).
Let's break down the costs:
1. Variable costs per [tex]$T$[/tex]-shirt:
- Cost per shirt: [tex]$\$[/tex]7[tex]$
- Cost for ink per shirt: $[/tex]\[tex]$2$[/tex]
- Cost for a bag per shirt: [tex]$\$[/tex]0.10[tex]$
Adding these together, the total variable cost per shirt is:
\[
7 + 2 + 0.10 = 9.10
\]
2. Fixed costs per month:
- Rent: $[/tex]\[tex]$500$[/tex]
- Electricity: [tex]$\$[/tex]40[tex]$
- Advertising: $[/tex]\[tex]$30$[/tex]
Adding these together, the total fixed costs per month are:
[tex]\[
500 + 40 + 30 = 570
\][/tex]
Now, the cost function, [tex]\( C \)[/tex], which gives the total monthly cost depending on the number of [tex]$T$[/tex]-shirts sold [tex]\( n \)[/tex], can be constructed as follows:
[tex]\[
C(t) = (\text{variable cost per shirt} \times \text{number of shirts}) + \text{fixed costs}
\][/tex]
[tex]\[
C(t) = (9.10n) + 570
\][/tex]
So, the correct answer is:
D. [tex]\( C = 9.10n + 570 \)[/tex]
