Final answer:
The average cost function A(x) for a landscaping company's lawn mowing service is obtained by dividing the total cost C(x) by the number of clients x, which is A(x) = (10000 + 3x + 0.01x^(5/2)) / x. This function helps find the optimal number of clients to minimize costs.
Step-by-step explanation:
To calculate the average cost of lawn mowing services at a landscaping company, we need to divide the total monthly cost by the number of clients. Given the total cost function C(x) = 10000 + 3x + 0.01x5/2, where x is the number of lawn mowing clients per month, the average cost function, A(x), is found by dividing C(x) by x.
The formula for the average cost function becomes A(x) = C(x) / x = (10000 + 3x + 0.01x5/2) / x. This equation represents how the cost per client can be optimized as the client base grows. Initially, the average cost is higher due to significant fixed costs, but as more clients are accepted (x increases), these costs are spread out, decreasing the average cost until it reaches a minimum. Beyond that point, due to diminishing returns, the average cost will start to increase again.
Therefore, the optimal average cost occurs at the output level where the average cost curve is at its minimum, before rising due to the impact of increasing variable costs and diminishing returns.