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Math  /  Algebra

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\$ 35 to set up their T-shirt printing machine with the design and $4.25\$ 4.25 in materials per T -shirt to print. 1) Create an equation to represent the average cost⁡C(x)\operatorname{cost} C(x), in dollars, per TT-shirt if xTx 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\$ 5 or less per shirt? Explain how you know.

Studdy Solution

STEP 1

1. The setup cost for the T-shirt printing machine is \$35.
2. The material cost per T-shirt is \$4.25.
3. \( x \) represents the number of T-shirts printed.

STEP 2

1. Create an equation for the average cost per T-shirt.
2. Calculate the average cost per shirt for 25 and 100 shirts.
3. Determine the minimum average cost per T-shirt.
4. Calculate the number of shirts needed for an average cost of \$5 or less.

STEP 3

To find the average cost per T-shirt, we need to account for both the setup cost and the material cost.
The total cost for printing x x T-shirts is: Total Cost=35+4.25x \text{Total Cost} = 35 + 4.25x
The average cost per T-shirt, C(x) C(x) , is given by dividing the total cost by the number of T-shirts: C(x)=35+4.25xx C(x) = \frac{35 + 4.25x}{x}

STEP 4

To find the average cost per shirt for 25 shirts, substitute x=25 x = 25 into the equation: C(25)=35+4.25×2525 C(25) = \frac{35 + 4.25 \times 25}{25}
Calculate: C(25)=35+106.2525=141.2525=5.65 C(25) = \frac{35 + 106.25}{25} = \frac{141.25}{25} = 5.65
To find the average cost per shirt for 100 shirts, substitute x=100 x = 100 into the equation: C(100)=35+4.25×100100 C(100) = \frac{35 + 4.25 \times 100}{100}
Calculate: C(100)=35+425100=460100=4.60 C(100) = \frac{35 + 425}{100} = \frac{460}{100} = 4.60

STEP 5

The average cost per T-shirt will decrease as more shirts are printed because the fixed setup cost is spread over more shirts.
The cheapest the average cost can get is the material cost per shirt, which is \$4.25, as the setup cost becomes negligible for a very large number of shirts.

STEP 6

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 x to clear the fraction: 35+4.25x≤5x 35 + 4.25x \leq 5x
Rearrange the inequality: 35≤0.75x 35 \leq 0.75x
Solve for x x : x≥350.75≈46.67 x \geq \frac{35}{0.75} \approx 46.67
Since x x must be a whole number, round up to the nearest whole number: x≥47 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)=35+4.25xx 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

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