C(x) = 90000 + 34x
R(x) = 413x
What is P(x)? Simplify your answer.
Answer (2)
Define the profit function as revenue minus cost: P ( x ) = R ( x ) − C ( x ) .
Substitute the given revenue and cost functions: P ( x ) = 413 x − ( 90000 + 34 x ) .
Simplify the expression: P ( x ) = 413 x − 90000 − 34 x .
Combine like terms to obtain the simplified profit function: P ( x ) = 379 x − 90000 .
The profit function is P ( x ) = 379 x − 90000 .
Explanation
Understanding the Problem We are given the cost function C ( x ) = 90000 + 34 x and the revenue function R ( x ) = 413 x . We need to find the profit function P ( x ) , which is the difference between the revenue and the cost.
Defining the Profit Function The profit function is defined as P ( x ) = R ( x ) − C ( x ) .
Substituting the Expressions Substitute the given expressions for R ( x ) and C ( x ) into the profit function: P ( x ) = 413 x − ( 90000 + 34 x )
Simplifying the Expression Simplify the expression by combining like terms: P ( x ) = 413 x − 90000 − 34 x
Combining Like Terms Further simplify to get: P ( x ) = ( 413 − 34 ) x − 90000
Calculating the Coefficient Calculate the coefficient of x :
413 − 34 = 379
Final Profit Function Write the final simplified profit function: P ( x ) = 379 x − 90000
Examples
Understanding profit functions is crucial in business. For example, if you're selling lemonade, C ( x ) could represent the cost of setting up your stand and the ingredients, while R ( x ) represents the money you make from selling each cup. The profit function P ( x ) helps you determine how many cups you need to sell to start making a profit and how much profit you'll make at different sales volumes. This concept is fundamental in making informed business decisions.
The profit function is calculated as the difference between revenue and cost, yielding the formula P(x) = 379x - 90000. This means that for each unit sold, the profit increases by 379, while there is a fixed cost of 90000 to cover. This function is crucial for understanding potential profitability based on sales volumes.
;
