The formulas below are the cost and revenue functions for a company that manufactures and sells small radios.
[tex]C(x)=24,000+40 x \text { and } R(x)=42 x[/tex]
a. Use the formulas shown to write the company's profit function, P, from producing and selling x radios.
b. Find the company's profit if 15,000 radios are produced and sold.
a. The company's profit function is [tex]P ( x )=[/tex] $\square$ (Simplify your answer.)
Answer (2)
Define the profit function as the difference between revenue and cost: P ( x ) = R ( x ) − C ( x ) .
Substitute the given revenue and cost functions: P ( x ) = 42 x − ( 24000 + 40 x ) .
Simplify the profit function: P ( x ) = 2 x − 24000 .
Evaluate the profit at x = 15000 : P ( 15000 ) = 2 ( 15000 ) − 24000 = 6000 . The company's profit is 6000 .
Explanation
Understanding Profit Function The profit function, P ( x ) , is the difference between the revenue function, R ( x ) , and the cost function, C ( x ) . In other words, the profit is what you get after subtracting the costs from the revenue.
Setting up the Profit Function We are given the revenue function R ( x ) = 42 x and the cost function C ( x ) = 24000 + 40 x . To find the profit function P ( x ) , we subtract the cost function from the revenue function:
P ( x ) = R ( x ) − C ( x )
Substituting Revenue and Cost Functions Now, substitute the given expressions for R ( x ) and C ( x ) :
P ( x ) = 42 x − ( 24000 + 40 x )
Simplifying the Profit Function Simplify the expression by distributing the negative sign and combining like terms:
P ( x ) = 42 x − 24000 − 40 x
P ( x ) = ( 42 x − 40 x ) − 24000
P ( x ) = 2 x − 24000
Evaluating Profit for 15,000 Radios Now we need to find the profit if 15,000 radios are produced and sold. This means we need to evaluate P ( 15000 ) .
P ( 15000 ) = 2 ( 15000 ) − 24000
Calculating the Profit Calculate the profit:
P ( 15000 ) = 30000 − 24000
P ( 15000 ) = 6000
Final Answer Therefore, the company's profit function is P ( x ) = 2 x − 24000 , and the company's profit if 15,000 radios are produced and sold is $6000.
Examples
Understanding profit functions is crucial for businesses. For instance, if you're running a lemonade stand, your revenue is the money you make from selling lemonade, and your costs include the lemons, sugar, and cups. The profit function helps you determine how many cups of lemonade you need to sell to cover your costs (break-even point) and start making a profit. By analyzing this function, you can make informed decisions about pricing, production levels, and cost management to maximize your earnings.
The company's profit function is P ( x ) = 2 x − 24000 . If 15,000 radios are produced and sold, the profit is 6000 dollars. This profit is calculated by evaluating the profit function at x = 15000 .
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