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The 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.

Create an equation to represent the average cost , in dollars, per T-shirt if T-shirts are printed by this business.
What is the average cost per shirt to print 25 shirts? 100 shirts?
What is the cheapest the average cost per T-shirt will get? Explain or show your reasoning.
How many shirts should be printed to have an average cost of $5 or less per shirt? Explain how you know.

User Styks
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The equation for the average cost per T-shirt (C) is:

C = (35 + 4.25T) / T

The average cost per shirt to print 25 shirts is $5.65.

The average cost per shirt to print 100 shirts is $4.60.

The cheapest average cost per T-shirt will occur when the number of shirts approaches infinity (n → ∞).

To have an average cost of $5 or less per shirt, you would need to print at least 47 shirts.

How to represent the average cost per T-shirt

To represent the average cost per T-shirt if T-shirts are printed by this business, consider the setup cost ($35) and the cost of materials per T-shirt ($4.25).

Let's denote the number of shirts as "T."

The equation for the average cost per T-shirt (C) is:

C = (35 + 4.25T) / T

To calculate the average cost per shirt to print 25 shirts, substitute n = 25 into the equation:

C = (35 + 4.25 * 25) / 25 = (35 + 106.25) / 25 = 141.25 / 25 = 5.65

The average cost per shirt to print 25 shirts is $5.65.

Similarly, to calculate the average cost per shirt to print 100 shirts:

C = (35 + 4.25 * 100) / 100 = (35 + 425) / 100 = 460 / 100 = 4.60

The average cost per shirt to print 100 shirts is $4.60.

To find the cheapest average cost per T-shirt, minimize the equation

C = (35 + 4.25T) / T.

Since the setup cost ($35) remains constant regardless of the number of shirts, the average cost per T-shirt will decrease as the number of shirts (n) increases.

Therefore, the cheapest average cost per T-shirt will occur when the number of shirts approaches infinity (n → ∞).

To determine how many shirts should be printed to have an average cost of $5 or less per shirt, we can set up the inequality:

5 ≥ (35 + 4.25n) / n

Multiplying both sides by n, we get:

5n ≥ 35 + 4.25n

0.75n ≥ 35

n ≥ 35 / 0.75

n ≥ 46.67

Therefore, to have an average cost of $5 or less per shirt, you would need to print at least 47 shirts.

User Kevin Sandow
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