Jenna is opening a clothing store. She plans to start by selling T-shirts. It costs her [tex]\$7[/tex] for each shirt, [tex]\$2[/tex] for ink per shirt, and [tex]\$0.10[/tex] per bag. Jenna also spends [tex]\$500[/tex] on rent, [tex]\[tex]$40[/tex] on electricity, and [tex]\$[/tex]30[/tex] on advertising each month. What is the cost function for Jenna's clothing store per month?

A. [tex]C = 9.10n + 570[/tex]

B. [tex]C = 7.00n + 500[/tex]

C. [tex]C = 570n + 9.10[/tex]

D. [tex]C = 7.00n + 570[/tex]



Answer :

Let's determine the cost function for Jenna's clothing store. The cost function is a mathematical expression that represents all costs incurred by Jenna, both fixed and variable, as a function of the number of T-shirts [tex]\(n\)[/tex] she sells.

1. Variable Costs:
- The cost per shirt is [tex]\(\$ 7\)[/tex].
- The cost for ink per shirt is [tex]\(\$ 2\)[/tex].
- The cost for a bag per shirt is [tex]\(\$ 0.10\)[/tex].

To find the total variable cost per shirt, we need to add these costs together:
[tex]\[ 7 + 2 + 0.10 = 9.10 \][/tex]
So, the variable cost per shirt is [tex]\(\$ 9.10\)[/tex].

2. Fixed Costs:
- The rent is [tex]\(\$ 500\)[/tex] per month.
- The electricity cost is [tex]\(\$ 40\)[/tex] per month.
- The advertising cost is [tex]\(\$ 30\)[/tex] per month.

Adding these fixed costs together, we get:
[tex]\[ 500 + 40 + 30 = 570 \][/tex]
So, the total fixed costs per month are [tex]\(\$ 570\)[/tex].

3. Cost Function:
The total cost function combines the fixed costs and the variable costs, which depend on the number of T-shirts sold. The cost function [tex]\(C\)[/tex] is given by:
[tex]\[ C = (\text{variable cost per shirt} \times n) + \text{fixed costs} \][/tex]
Substituting in the values we calculated, the cost function is:
[tex]\[ C = 9.10n + 570 \][/tex]

Therefore, the correct cost function for Jenna's clothing store per month is:
[tex]\[ \boxed{C = 9.10n + 570} \][/tex]

So, the correct answer is A. [tex]\(\mathbf{C=9.10n + 570}\)[/tex].