The revenue (in millions of dollars) from the sale of [tex]$x$[/tex] units at a home supply outlet is given by [tex]$R(x)=0.12 x$[/tex]. The profit (in millions of dollars) from the sale of [tex]$x$[/tex] units is given by [tex]$P ( x )=0.085 x -0.9$[/tex].
What is the cost of producing 8 units?
(Please provide your answer in MILLIONS)
Answer (1)
Define the cost function as the difference between revenue and profit: C ( x ) = R ( x ) − P ( x ) .
Substitute the given revenue and profit functions: C ( x ) = 0.12 x − ( 0.085 x − 0.9 ) .
Simplify the cost function: C ( x ) = 0.035 x + 0.9 .
Evaluate the cost function at x = 8 : C ( 8 ) = 0.035 ( 8 ) + 0.9 = 1.18 . The cost of producing 8 units is 1.18 .
Explanation
Understanding the Problem We are given the revenue function R ( x ) = 0.12 x and the profit function P ( x ) = 0.085 x − 0.9 , both in millions of dollars, where x is the number of units sold. We want to find the cost of producing 8 units.
Relating Cost, Revenue, and Profit We know that Profit = Revenue - Cost, which can be rearranged to Cost = Revenue - Profit. Therefore, to find the cost function C ( x ) , we subtract the profit function from the revenue function: C ( x ) = R ( x ) − P ( x ) .
Substituting the Functions Substitute the given functions for R ( x ) and P ( x ) : C ( x ) = ( 0.12 x ) − ( 0.085 x − 0.9 ) .
Simplifying the Cost Function Simplify the expression for C ( x ) : C ( x ) = 0.12 x − 0.085 x + 0.9 = 0.035 x + 0.9 .
Evaluating C(8) Now, we want to find the cost of producing 8 units, so we need to evaluate C ( 8 ) : C ( 8 ) = 0.035 ( 8 ) + 0.9 .
Calculating the Cost Calculate C ( 8 ) : C ( 8 ) = 0.28 + 0.9 = 1.18 .
Final Answer The cost of producing 8 units is 1.18 million dollars.
Examples
Understanding cost, revenue, and profit is crucial in business. For example, if a company sells handmade sweaters, they can use these functions to determine the cost of producing a certain number of sweaters, given the revenue they generate and the profit they make. By analyzing these relationships, the company can make informed decisions about pricing, production levels, and overall business strategy. This helps in optimizing their operations and maximizing profitability. For instance, if the revenue from selling 100 sweaters is R ( 100 ) = 0.12 × 100 = 12 million dollars and the profit is P ( 100 ) = 0.085 × 100 − 0.9 = 7.6 million dollars, then the cost of producing 100 sweaters is C ( 100 ) = R ( 100 ) − P ( 100 ) = 12 − 7.6 = 4.4 million dollars.
