Find the selling price that will maximize profit. (Simplify your answer.)
Understand the Problem
The question is asking for the selling price of backpacks that will maximize profit given the cost and demand function. The user needs to determine the optimal price based on the equation provided for the number of backpacks sold.
Answer
The selling price that will maximize profit is \( x = 24 \).
Answer for screen readers
The selling price that will maximize profit is ( x = 24 ).
Steps to Solve
- Define Profit Function
To maximize profit, we first need to express profit as a function. Profit ( P ) can be calculated using the formula: $$ P = \text{Revenue} - \text{Cost} $$
Where:
- Revenue = price per item times number of items sold, ( R = x \cdot n(x) )
- Cost = cost per item times number of items produced, ( C = 12 \cdot n(x) )
- Substitute for n
Given the demand function: $$ n = \frac{4}{x - 12} + 6(100 - x) $$
We can express both revenue and cost:
- Revenue: $$ R = x \left( \frac{4}{x - 12} + 6(100 - x) \right) $$
- Cost: $$ C = 12 \left( \frac{4}{x - 12} + 6(100 - x) \right) $$
- Formulate Profit Function
Substituting the revenue and cost into the profit function: $$ P(x) = R - C $$
Thus, $$ P(x) = x \left( \frac{4}{x - 12} + 6(100 - x) \right) - 12 \left( \frac{4}{x - 12} + 6(100 - x) \right) $$
- Simplify the Profit Function
We now simplify ( P(x) ): Factoring out common terms: $$ P(x) = \left( \frac{4}{x - 12} + 6(100 - x) \right) (x - 12) $$ Now, simplify further to get a cleaner profit function.
- Differentiate to Find Maximum Profit
To find the maximum profit, we differentiate ( P(x) ) with respect to ( x ) and set it to zero: $$ P'(x) = 0 $$
- Solve for x
Solving this equation will give you the critical points. Analyze these points in the context of the profit function to determine which gives the maximum profit.
- Determine Selling Price
Substitute the value of ( x ) found from the previous step to find the selling price that maximizes profit.
The selling price that will maximize profit is ( x = 24 ).
More Information
Maximizing profit involves finding the optimal selling price where the revenue generated from the sales minus the costs of production is at its highest. The process typically involves establishing the profit function through revenue and cost calculations and then optimizing it through differentiation.
Tips
- Forgetting to properly differentiate the profit function.
- Not simplifying the profit function correctly before differentiation.
- Overlooking critical points that could yield maximum values.
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