Understanding Profit Function: Calculating Total Profit for a Scooter Company

For a scooter company determining overall profitability, it's critical to establish a profit function that accounts for both production costs and revenue. In this context, we understand that the production cost for each scooter is $200, and the company shoulders a fixed cost of $1,500. Total revenue is indicated by the function $R(x) = 400x - 2x^2$, where $x$ represents the number of electric scooters sold.

The profit function, which we aim to outline, is essentially the total revenue minus the total cost. The total cost not only includes the variable costs per unit produced but also the fixed costs irrespective of the number of units produced.

Given the cost per scooter and fixed costs, the total cost function $C(x)$ can be expressed as:

$$C(x) = 200x + 1500$$

Now, to determine the profit function $P(x)$, we subtract this total cost function from the revenue function:

$$P(x) = R(x) - C(x)$$

Substituting in the given functions:

$$P(x) = (400x - 2x^2) - (200x + 1500)$$

Simplifying, we find the profit function:

$$P(x) = -2x^2 + 200x - 1500$$

This quadratic profit function represents the scooter company's total profit and is an essential tool in forecasting financial outcomes and planning for scalable growth. By analyzing this function, the company can identify the break-even point, optimize pricing, determine the best production levels, and make informed strategic decisions to increase profitability.

For those looking to dive deeper into cost management and profit optimization strategies for businesses, understanding and utilizing the profit function is a cornerstone of financial analysis and business administration.