The revenue function of a business is $R(x)=50x$ where $x$ is the number of units sold. The expenditure function is $E(x)=40x+200$.
a) Write the expression for the profit function. Let $P(x)$ be the profit function.
b) If 100 items were sold, what is the profit?
Answer (1)
Define the profit function as the difference between revenue and expenditure: P ( x ) = R ( x ) − E ( x ) .
Substitute the given revenue and expenditure functions: P ( x ) = 50 x − ( 40 x + 200 ) .
Simplify the profit function: P ( x ) = 10 x − 200 .
Calculate the profit for 100 items: P ( 100 ) = 10 ( 100 ) − 200 = 800 . The profit is 800 .
Explanation
Understanding the Problem We are given the revenue function R ( x ) = 50 x and the expenditure function E ( x ) = 40 x + 200 , where x is the number of units sold. We need to find the profit function P ( x ) and then calculate the profit when 100 items are sold.
Finding the Profit Function The profit function, P ( x ) , is the difference between the revenue function, R ( x ) , and the expenditure function, E ( x ) . Therefore, we have P ( x ) = R ( x ) − E ( x ) Substituting the given expressions for R ( x ) and E ( x ) , we get P ( x ) = 50 x − ( 40 x + 200 ) Simplifying the expression, we have P ( x ) = 50 x − 40 x − 200 P ( x ) = 10 x − 200
Calculating the Profit Now, we need to find the profit when 100 items are sold. This means we need to evaluate P ( 100 ) . Substituting x = 100 into the profit function, we get P ( 100 ) = 10 ( 100 ) − 200 P ( 100 ) = 1000 − 200 P ( 100 ) = 800
Examples
Understanding profit functions is crucial in business. For instance, if you're selling lemonade, your revenue is the money you make per cup, and your expenditure includes the cost of lemons, sugar, and cups. The profit function helps you determine how many cups you need to sell to cover your costs and start making a profit. By analyzing this function, you can make informed decisions about pricing and production levels to maximize your earnings. This concept applies to businesses of all sizes, from lemonade stands to large corporations.
