C(x) = 98,020 + 89x(x+1)
R(x) = 46x
What is company profit P(x) = ?
Answer (2)
The profit function is defined as the revenue function minus the cost function: P ( x ) = R ( x ) − C ( x ) .
Substitute the given revenue and cost functions: P ( x ) = 46 x − ( 98020 + 89 x ( x + 1 )) .
Simplify the expression: P ( x ) = 46 x − 98020 − 89 x 2 − 89 x .
Combine like terms to find the profit function: P ( x ) = − 89 x 2 − 43 x − 98020 . The company profit function is P ( x ) = − 89 x 2 − 43 x − 98020 .
Explanation
Understanding the Problem We are given the cost function C ( x ) = 98020 + 89 x ( x + 1 ) and the revenue function R ( x ) = 46 x . We want to find the profit function P ( x ) .
Defining the Profit Function The profit function is the difference between the revenue and the cost, so 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 ) = 46 x − ( 98020 + 89 x ( x + 1 )) .
Simplifying the Expression Now, simplify the expression for P ( x ) : P ( x ) = 46 x − ( 98020 + 89 x 2 + 89 x ) P ( x ) = 46 x − 98020 − 89 x 2 − 89 x
Combining Like Terms Combine like terms to obtain the final expression for P ( x ) : P ( x ) = − 89 x 2 + ( 46 x − 89 x ) − 98020 P ( x ) = − 89 x 2 − 43 x − 98020
Final Answer Therefore, the profit function is P ( x ) = − 89 x 2 − 43 x − 98020 .
Examples
Understanding profit functions is crucial for businesses. For example, a small business owner can use the profit function to determine the number of units they need to sell to break even or to maximize their profit. By analyzing the profit function, they can make informed decisions about pricing, production levels, and cost management. For instance, if the profit function is P ( x ) = − 2 x 2 + 100 x − 500 , the business owner can find the production level x that maximizes profit by finding the vertex of the quadratic function. In this case, the vertex occurs at x = 2 a − b = 2 ( − 2 ) − 100 = 25 . So, producing 25 units would maximize profit.
The profit function can be calculated using the formula P ( x ) = R ( x ) − C ( x ) . Substituting the provided revenue and cost functions, we find that P ( x ) = − 89 x 2 − 43 x − 98 , 020 . This expression helps analyze how profit varies with the number of units sold.
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