一位女售货员推销吸尘器，利润方程式是什么？

A saleswoman is selling vacuum cleaners door to door. She is able to sell each one for $75. She spent $1500 on her vacuum cleaner inventory, and an additional $8 per vacuum on bags. What is the profit equation?Each vacuum cleaner sold yields a direct revenue of $75. However, to determine the true profit per unit, we must consider both the initial inventory cost and the additional expense per vacuum for bags. The saleswoman invested $1500 in acquiring the vacuum cleaners. Let's denote the total number of vacuum cleaners she bought as $N$. Since the total inventory cost is $1500, the cost per vacuum cleaner can be calculated as:$$\text{Cost per vacuum cleaner} = \frac{1500}{N}$$Furthermore, she sps an extra $8 on bags for each vacuum cleaner sold. Therefore, the total cost per vacuum cleaner, including both the inventory and bag costs, is:$$\text{Total cost per vacuum cleaner} = \frac{1500}{N} + 8$$Profit, by definition, is the revenue minus the cost. Hence, the profit equation for each vacuum cleaner sold is:$$\text{Profit per vacuum cleaner} = 75 - \left(\frac{1500}{N} + 8\right)$$Simplifying this equation, we get:$$\text{Profit per vacuum cleaner} = 67 - \frac{1500}{N}$$This equation illustrates how the profit varies based on the number of vacuum cleaners sold, $N$. As $N$ increases, the average inventory cost per unit decreases, leading to higher profits per vacuum cleaner sold.

一位女售货员正在挨家挨户推销吸尘器。她能以75美元的价格卖掉每一个。她在吸尘器库存上花费了1500美元，每台吸尘器在袋子上额外花费了8美元。利润方程式是什么？为了计算出这位女售货员的利润方程式，我们需要先确定几个关键变量和它们之间的关系。设她卖出的吸尘器数量为x台。那么，她的总收入将是75美元乘以x，即75x美元。接下来，我们计算她的总成本。她在吸尘器库存上的初始花费是1500美元，这是一个固定成本。每台吸尘器额外花费的8美元袋子费用则是变动成本，与售出的吸尘器数量成正比。因此，总变动成本是8x美元。将固定成本和变动成本相加，我们得到总成本方程：1500 + 8x美元。利润则是总收入减去总成本。所以，利润方程式为：利润 = 总收入 - 总成本= 75x - (1500 + 8x)= 75x - 1500 - 8x= 67x - 1500这个方程式告诉我们，女售货员的利润与她卖出的吸尘器数量成正比，但减去了一个固定的初始投资成本。为了开始盈利，她需要卖出足够多的吸尘器，使得67x超过1500。换句话说，只有当x大于约22.39（即1500除以67）时，她才能实现正的利润。随着x的增加，利润将迅速增长，因为每台吸尘器的边际利润（即每台吸尘器的售价减去额外成本）是67美元。