Unraveling the Profit Function of The Scooter Company's Electric Ventures

When it comes to understanding the profitability of a product or service, few things are as crucial as the profit function. For The Scooter Company, a pioneer in electric transportation, discerning the total profit from their electric scooter sales is imperative to gauging success. Costs and revenues are the cornerstones of this calculation; specifically, The Scooter Company's cost structure includes both variable and fixed costs, with each electric scooter costing $200 to produce and fixed costs amounting to $1,500. Revenue, on the other hand, is expressed by the function R(x) = 400x - 2x^2, where x represents the number of scooters sold.

To determine the total profit function, we must consider the cost of producing x scooters along with the fixed costs and then subtract this total cost from the revenue function. The cost of producing x scooters is represented by C(x) = 200x (since each scooter costs $200 to produce), and when added to the fixed costs, the total cost function T(x) becomes T(x) = 200x + 1,500.

Subtracting this total cost function from the revenue function gives us the profit function, P(x):

P(x) = R(x) - T(x)
= (400x - 2x^2) - (200x + 1,500)
= 400x - 2x^2 - 200x - 1,500
= -2x^2 + 200x - 1,500

The profit function P(x) = -2x^2 + 200x - 1,500 accurately represents the total profit for The Scooter Company. By analyzing this function, the company can understand how many units need to be sold to break even or achieve certain profit goals, a vital component of strategic business planning.

Calculating total profits is not only essential for internal assessments but also serves as a valuable metric for investors and stakeholders interested in the sustainability and growth potential of The Scooter Company's electric scooters. Understanding and projecting profits allow businesses to adjust strategies, optimize production, and potentially expand market reach to ensure long-term viability in the competitive electric vehicle industry.