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Understanding the Scooter Company's Total Profit Function
To determine the total profit for the scooter company, we need to account for both the revenue and the costs involved. Let's break this down step-by-step to obtain a comprehensive understanding of the profit function.
Analyzing Total Revenue and Costs
Revenue Function
The scooter company's revenue function is given by:
r(x) = 400x − 2x^2
where x represents the number of electric scooters sold.
Cost Analysis
- Cost to Produce Each Scooter: $150
- Fixed Costs: $2,000
Thus, the total cost C(x) of producing x scooters is:
C(x) = 150x + 2,000
Profit Function
Profit is defined as total revenue minus total costs. Therefore, the profit function P(x) can be represented as:
P(x) = r(x) - C(x)
Let's substitute the given equations:
P(x) = (400x - 2x^2) - (150x + 2,000)
Simplify this equation step-by-step:
P(x) = 400x - 2x^2 - 150x - 2,000
P(x) = -2x^2 + (400x - 150x) - 2,000
P(x) = -2x^2 + 250x - 2,000
Therefore, the correct function representing the scooter company's total profit is:
-2x^2 + 250x - 2,000
Conclusion
By breaking down the revenue and cost functions step-by-step, we've derived that the scooter company's total profit function is -2x^2 + 250x - 2,000. This function helps in making informed decisions based on the number of scooters sold and the associated profits.
