The Scooter Company manufactures and sells electric scooters. Each scooter costs $[tex]$200$[/tex] to manufacture. The revenue can be determined by the function [tex]$R(x)=300 x-3 x^2$[/tex], where [tex]$x$[/tex] represents each electric scooter sold. Which of the following represents the profit?
Answer (2)
Define the cost function: C ( x ) = 200 x .
Define the profit function: P ( x ) = R ( x ) − C ( x ) .
Substitute the revenue and cost functions: P ( x ) = ( 300 x − 3 x 2 ) − 200 x .
Simplify the profit function: P ( x ) = − 3 x 2 + 100 x .
Explanation
Understanding the Problem The problem provides the revenue function R ( x ) = 300 x − 3 x 2 and states that each scooter costs $200 to manufacture. We need to find the profit function, which is the revenue minus the cost.
Defining the Cost Function First, we need to define the cost function. Since each scooter costs 200 t o man u f a c t u re , t h ecos t f u n c t i o n C(x) f or x scoo t ers i s g i v e nb y : C ( x ) = 200 x $
Setting up the Profit Function The profit function P ( x ) is the revenue function minus the cost function: P ( x ) = R ( x ) − C ( x ) Substituting the given revenue function and the cost function, we get: P ( x ) = ( 300 x − 3 x 2 ) − ( 200 x ) P ( x ) = 300 x − 3 x 2 − 200 x
Simplifying the Profit Function Now, we simplify the profit function: P ( x ) = − 3 x 2 + ( 300 x − 200 x ) P ( x ) = − 3 x 2 + 100 x
Finding the Correct Option Comparing the simplified profit function P ( x ) = − 3 x 2 + 100 x with the multiple-choice options, we find that the correct answer is: − 3 x 2 + 100 x − 1200 is not the profit function. − 3 x 2 + 200 x − 1200 is not the profit function. − 3 x 2 − 200 x − 1500 is not the profit function. − 600 x 3 − 4500 x 2 + 60000 x + 450000 is not the profit function. However, the profit function we derived, P ( x ) = − 3 x 2 + 100 x , is not exactly present in the options. Let's re-examine the question and the options. It seems there might be a constant term missing in our calculation or in the provided revenue function. If there was a fixed cost of 1200 , t h e n t h e p ro f i t f u n c t i o n w o u l d b e P(x) = -3x^2 + 100x - 1200 . Ho w e v er , ba se d o n t h e in f or ma t i o n g i v e n , t h ecorrec tp ro f i t f u n c t i o ni s : P ( x ) = − 3 x 2 + 100 x $
Final Answer Since the derived profit function P ( x ) = − 3 x 2 + 100 x is not directly available in the options, and assuming there might be a typo in the options or missing information in the problem statement, we choose the closest option. If we assume a fixed cost of 1200 , t h e p ro f i t f u n c t i o n w o u l d b e P(x) = -3x^2 + 100x - 1200 . Ho w e v er , w i t h o u tt hi s in f or ma t i o n , t h e m os t a cc u r a t ere p rese n t a t i o n o f t h e p ro f i t ba se d o n t h e g i v e nin f or ma t i o ni s : P ( x ) = − 3 x 2 + 100 x $
Examples
Understanding profit functions is crucial for businesses. For example, a local bakery can use a profit function to determine how many cakes they need to sell each day to maximize their profit. By analyzing the cost of ingredients, the selling price of each cake, and other operational costs, they can create a profit function. This function helps them make informed decisions about pricing, production levels, and overall business strategy, ensuring they operate efficiently and profitably.
The profit function for the Scooter Company is given by P ( x ) = − 3 x 2 + 100 x , where x represents the number of electric scooters sold. This function accounts for both revenue and manufacturing costs. The profit increases with sales until a certain level, reflecting the cost dynamics of the business.
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