Question

The school band is selling t-shirts for a fundraiser. The initial cost for a steamer to apply the decals is $100. It is predicted that the cost of material for each shirt is $12. The function that shows the band’s average cost per t-shirt after x t-shirts are sold is:

f(x) =
100 + 12x
x

Answer 1

First rewrite the function:

f(x) = (100 + 12x)/x = 12 + 100/x.

Interpretation:
- 12 is the variable (per-shirt) cost.
- 100/x is the fixed steamer cost spread over x shirts.
- Domain: x > 0 (in practice x is a positive integer).

Behavior:
- As x increases, 100/x → 0, so f(x) → 12. The average cost per shirt approaches $12.
- f(1) = 112 (one shirt is expensive), f(10) = 22, f(50) = 14, f(100) = 13.

To find how many shirts are needed to reach a target average cost C (>12):
12 + 100/x ≤ C ⟹ x ≥ 100/(C − 12).
Example: to get average ≤ $13 you need x ≥ 100/(1) = 100 shirts (exactly 100 gives $13; more than 100 gives less than $13).