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.