The revenue from selling $x$ shirts is $r(x)=15x$.
The cost of buying $x$ shirts is $c(x)=7x+20$.
The profit from selling $x$ shirts is $p(x)=r(x)-c(x)$.
What is $p(x)$?
A. $p(x)=22x-20$
B. $p(x)=8x+20$
C. $p(x)=8x-20$
D. $p(x)=22x+20
Answer (2)
Substitute the expressions for r ( x ) and c ( x ) into the equation p ( x ) = r ( x ) − c ( x ) .
Simplify the expression: p ( x ) = 15 x − ( 7 x + 20 ) = 15 x − 7 x − 20 .
Combine like terms: p ( x ) = 8 x − 20 .
The profit function is 8 x − 20 .
Explanation
Understanding the Problem We are given the revenue function r ( x ) = 15 x and the cost function c ( x ) = 7 x + 20 . The profit function is defined as p ( x ) = r ( x ) − c ( x ) . We need to find the expression for p ( x ) .
Calculating the Profit Function To find p ( x ) , we substitute the given expressions for r ( x ) and c ( x ) into the equation p ( x ) = r ( x ) − c ( x ) . So, we have p ( x ) = ( 15 x ) − ( 7 x + 20 ) p ( x ) = 15 x − 7 x − 20 p ( x ) = ( 15 − 7 ) x − 20 p ( x ) = 8 x − 20
Comparing with Options Now we compare our expression for p ( x ) with the given options. We found that p ( x ) = 8 x − 20 , which matches option C.
Final Answer Therefore, the profit function is p ( x ) = 8 x − 20 .
Examples
Understanding profit functions is crucial in business. For example, if you're selling lemonade, knowing your revenue (money from sales) and costs (ingredients, cups) helps you calculate your profit. If each cup of lemonade sells for $1.50 and costs $0.50 to make, plus a fixed cost of 5 f or t h es t an d , yo u r p ro f i t f u n c t i o n w o u l d b e p(x) = (1.50 - 0.50)x - 5 = x - 5 , w h ere x$ is the number of cups sold. This helps you determine how many cups you need to sell to break even or reach a desired profit.
The profit function is calculated by subtracting the cost function from the revenue function, resulting in p ( x ) = 8 x − 20 . This matches option C from the provided choices. Thus, the correct answer is C.
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