A company can produce T-shirts for $0.99 each. They spent $1,000,000 on their building and machines. What is the horizontal asymptotes of the graph that models the average cost of each t shirt, including the start up cost?
total cost for x shirts is
c(x) = 1000000 + .99x
average cost is
a(x) = c(x)/x = .99 + 1000000/x
asymptote at a = .99
makes sense, since for say, a billion shirts, they will each cost .99 + .001
(the fixed cost is shared among a billion shirts). The average cost will never get below .99, but will get closer and closer to it, as the fixed amount is spread among more and more shirts.
To find the horizontal asymptote of the graph that models the average cost of each T-shirt, including the startup cost, we need to consider the relationship between the average cost and the number of T-shirts produced.
The cost of each T-shirt can be calculated by dividing the total cost by the number of T-shirts produced. In this case, the startup cost of $1,000,000 should be spread out over all the T-shirts produced.
Let's denote the average cost of each T-shirt as C and the number of T-shirts produced as x. The cost equation can be written as:
C = (1,000,000 + 0.99x) / x
As x approaches infinity (i.e., producing an infinite number of T-shirts), the startup cost becomes less significant in comparison to the total cost. Therefore, the average cost tends to approach the variable cost per T-shirt, which in this case is $0.99.
The horizontal asymptote represents the value that the average cost approaches as x becomes large. In this case, the horizontal asymptote is $0.99.
So, the horizontal asymptote of the graph that models the average cost of each T-shirt, including the startup cost, is $0.99.