The formulas below are the cost and revenue functions for a company that manufactures and sells small radios.
[tex]C(x)=90,000+34 x \text { and } R(x)=43 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 24,000 radios are produced and sold.
a. The company's profit function is [tex]P ( x )=9 x -90,000[/tex]. (Simplify your answer.)
b. The company's profit from selling 24,000 radios is [tex]$\square[/tex]. (Simplify your answer.)
Answer (2)
Determine the profit function by subtracting the cost function from the revenue function: P ( x ) = R ( x ) − C ( x ) = 43 x − ( 90000 + 34 x ) = 9 x − 90000 .
Substitute x = 24000 into the profit function: P ( 24000 ) = 9 ( 24000 ) − 90000 .
Calculate the profit: P ( 24000 ) = 216000 − 90000 = 126000 .
The company's profit from selling 24,000 radios is 126000 .
Explanation
Problem Analysis Let's analyze the given problem. We have the cost function C ( x ) = 90000 + 34 x and the revenue function R ( x ) = 43 x . The profit function, P ( x ) , is the difference between the revenue and the cost, i.e., P ( x ) = R ( x ) − C ( x ) . We need to find the profit function and then calculate the profit when x = 24000 radios are produced and sold.
Finding the Profit Function First, we need to find the profit function P ( x ) . We know that P ( x ) = R ( x ) − C ( x ) . Substituting the given functions, we have: P ( x ) = 43 x − ( 90000 + 34 x ) P ( x ) = 43 x − 90000 − 34 x P ( x ) = ( 43 x − 34 x ) − 90000 P ( x ) = 9 x − 90000 So, the profit function is P ( x ) = 9 x − 90000 .
Calculating the Profit Now, we need to find the profit when 24,000 radios are produced and sold. We substitute x = 24000 into the profit function: P ( 24000 ) = 9 ( 24000 ) − 90000 P ( 24000 ) = 216000 − 90000 P ( 24000 ) = 126000 Therefore, the company's profit from selling 24,000 radios is $126 , 000 .
Examples
Understanding cost, revenue, and profit functions is crucial for businesses to make informed decisions. For example, a small bakery can use these functions to determine the optimal number of cakes to bake and sell to maximize their profit. By analyzing their fixed costs (rent, utilities) and variable costs (ingredients, labor), they can create cost and revenue functions. The profit function then helps them identify the break-even point and the production level that yields the highest profit. This approach ensures the bakery operates efficiently and profitably.
The company's profit function is P ( x ) = 9 x − 90 , 000 . When 24,000 radios are sold, the profit is $126,000. This is calculated using the profit formula and substituting x = 24000 into it.
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