The formulas below are the cost and revenue functions for a company that manufactures and sells small radios.
[tex]C(x)=98,000+39 x \text { and } R(x)=46 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 18,000 radios are produced and sold.
a. The company's profit function is [tex]P ( x )=7 x -98,000[/tex]. (Simplify your answer.)
b. The company's profit from selling 18,000 radios is $ [$\square$] . (Simplify your answer.)
Answer (1)
Substitute x = 18000 into the profit function P ( x ) = 7 x − 98000 .
Calculate P ( 18000 ) = 7 ( 18000 ) − 98000 .
Simplify the expression to find the profit: 126000 − 98000 = 28000 .
The company's profit from selling 18,000 radios is 28000 .
Explanation
Understanding the Problem We are given the profit function P ( x ) = 7 x − 98000 , where x is the number of radios produced and sold. We need to find the profit when 18,000 radios are produced and sold.
Substituting the Value of x To find the profit from selling 18,000 radios, we substitute x = 18000 into the profit function: P ( 18000 ) = 7 ( 18000 ) − 98000
Calculating the Profit Now, we calculate the profit: P ( 18000 ) = 7 × 18000 − 98000 = 126000 − 98000 = 28000 So, the company's profit from selling 18,000 radios is $28,000.
Final Answer Therefore, the company's profit from selling 18,000 radios is 28000 .
Examples
Understanding profit functions is crucial in business. For example, if you're selling lemonade, you can use a profit function to determine how many cups you need to sell to cover your costs (like lemons, sugar, and cups) and start making a profit. By analyzing the profit function, you can make informed decisions about pricing, production levels, and cost management to maximize your earnings. This helps in setting realistic goals and making strategic choices to ensure your business is successful and sustainable.
