QuestionThe school art club at a large high school is in charge of designing school T-shirts and getting them printed this year. A local business charges $\$ 35$ to set up their T-shirt printing machine with the design and $\$ 4.25$ in materials per T -shirt to print. 1) Create an equation to represent the average $\operatorname{cost} C(x)$, in dollars, per $T$-shirt if $x T$ shirts are printed by this business. 2) What is the average cost per shirt to print 25 shirts? 100 shirts? 3) What is the cheapest the average cost per T-shirt will get? Explain or show your reasoning. 4) How many shirts should be printed to have an average cost of $\$ 5$ or less per shirt? Explain how you know.

Studdy Solution
To find how many shirts should be printed to have an average cost of \$5 or less, set up the inequality: \[ C(x) = \frac{35 + 4.25x}{x} \leq 5 \]
Multiply both sides by $x$ to clear the fraction: $$35 + 4.25x \leq 5x$$
Rearrange the inequality: $$35 \leq 0.75x$$
Solve for $x$: $$x \geq \frac{35}{0.75} \approx 46.67$$
Since $x$ must be a whole number, round up to the nearest whole number: $$x \geq 47$$
Therefore, at least 47 shirts need to be printed to achieve an average cost of \$5 or less per shirt.
The solutions are: 1) $C(x) = \frac{35 + 4.25x}{x}$ 2) For 25 shirts: \$5.65; for 100 shirts: \$4.60 3) The cheapest average cost per T-shirt is \$4.25 4) At least 47 shirts need to be printed