Question

The manufacturer of the water toy "silly soaker" quotes a variable cost of $7.25 per unit and fixed costs of $9,000. create a function to represent the average cost per unit to manufacture the silly soaker, and use that model to determine the average cost per unit for a level of production of 푥 = 6,000 units. what is the horizontal asymptote of this function, and what does it represent?

Answer 1

The average cost function for the "silly soaker" is A(x) = (9000 + 7.25x) / x, with an average cost of $8.75 per unit at 6,000 units. The horizontal asymptote of this function is y = 7.25, representing the variable cost per unit.

Step-by-step explanation:

The function to represent the average cost per unit to manufacture the "silly soaker" is derived by adding the fixed costs to the variable costs (which is the variable cost per unit multiplied by the number of units) and dividing by the number of units. Mathematically, this can be written as A(x) = (Fixed Costs + (Variable Cost per Unit × x)) / x. In this case, the average cost function would be A(x) = (9000 + 7.25x) / x. Calculating the average cost for x = 6,000 units: A(6000) = (9000 + 7.25×6000) / 6000. Simplifying this, we get A(6000) = (9000 + 43500) / 6000, which equals 7.25 + 1.50 = 8.75. Therefore, the average cost per unit at 6,000 units is $8.75. The horizontal asymptote of this function is the line y = 7.25. The horizontal asymptote represents the variable cost per unit, which is also the minimum average cost as production goes to infinity.