Assume that the following has a linear cost function.
| Fixed Cost | Marginal Cost per item | Item Sells For |
|---|---|---|
| $850 | $9 | $35 |
Find the following.
The profit function is [tex]P(x) =[/tex]
(On this one, only give me the SLOPE of the PROFIT function.)
Answer (1)
Define the cost function: C ( x ) = 850 + 9 x .
Define the revenue function: R ( x ) = 35 x .
Define the profit function: P ( x ) = R ( x ) − C ( x ) = 26 x − 850 .
Identify the slope of the profit function: 26 .
Explanation
Problem Analysis Let's analyze the given information to determine the slope of the profit function. We have the fixed cost, the marginal cost per item, and the selling price per item. We'll use these to construct the cost, revenue, and profit functions.
Cost Function Let x be the number of items produced and sold. The cost function, C ( x ) , is the sum of the fixed cost and the variable cost (marginal cost per item times the number of items). So, we have: C ( x ) = 850 + 9 x
Revenue Function The revenue function, R ( x ) , is the selling price per item times the number of items sold. So, we have: R ( x ) = 35 x
Profit Function The profit function, P ( x ) , is the difference between the revenue and the cost: P ( x ) = R ( x ) − C ( x ) Substituting the expressions for R ( x ) and C ( x ) , we get: P ( x ) = 35 x − ( 850 + 9 x ) Simplifying, we have: P ( x ) = 35 x − 850 − 9 x = 26 x − 850
Slope of Profit Function The profit function is P ( x ) = 26 x − 850 . The slope of this linear function is the coefficient of x , which is 26. Therefore, the slope of the profit function is 26.
Interpretation of the Slope The slope of the profit function represents the change in profit for each additional item sold. In this case, for each additional item sold, the profit increases by $26.
Examples
Understanding profit functions is crucial in business. For example, if you're running a lemonade stand, the fixed cost might be the cost of the stand itself and the pitcher, while the marginal cost is the cost of lemons and sugar per cup. Knowing your profit function helps you determine how many cups you need to sell to break even or reach a certain profit goal. By analyzing the slope, you can quickly assess how much each additional sale contributes to your overall profit, guiding decisions on pricing and production levels.
