when an auto dealers mark up over factory price is x dollars per care he dealer sells (300-2/3 x)cars per month
what value of x will the profit be the greatest?
To find the value of x that will maximize the profit, we need to understand the relationship between the mark-up price, the number of cars sold per month, and the profit.
Let's break down the problem step by step:
1. Start with the formula for profit: Profit = (Selling Price - Factory Price) * Number of cars sold
2. In this case, the Selling Price is the Factory Price plus the mark-up price: Selling Price = Factory Price + Mark-up Price
3. Substitute the formula for the Selling Price into the Profit formula: Profit = (Factory Price + Mark-up Price - Factory Price) * Number of cars sold
4. Simplify the expression: Profit = Mark-up Price * Number of cars sold
Now that we have the formula for profit, we can determine the value of x that will maximize it.
5. Given that the dealer sells (300 - (2/3)x) cars per month, we can substitute this expression for the Number of cars sold: Profit = Mark-up Price * (300 - (2/3)x)
To maximize the profit, we'll need to differentiate the Profit formula with respect to x and find the critical points (where the derivative equals 0).
6. Differentiate the Profit formula with respect to x: d(Profit)/dx = d(Mark-up Price * (300 - (2/3)x))/dx
7. Simplify the derivative: d(Profit)/dx = -2/3 * Mark-up Price
Setting the derivative equal to 0 gives us the critical point:
-2/3 * Mark-up Price = 0
8. Solve for Mark-up Price: Mark-up Price = 0
Since the mark-up price cannot be 0 (as it would result in no profit), there is no critical point where the derivative equals zero.
Therefore, there is no specific value of x that will maximize the profit in this scenario.