Let [tex]$C(x)=1000+8 x+0.04 x^2$[/tex] be the cost function and [tex]$p(x)=32 x$[/tex] be the price function. Determine the production level that will maximize the profit.
a) 1000
b) 300
c) 200
d) 500
Answer (1)
Define the revenue function as R ( x ) = 32 x .
Define the profit function as P ( x ) = R ( x ) − C ( x ) = − 0.04 x 2 + 24 x − 1000 .
Find the critical points by taking the derivative of the profit function and setting it to zero: P ′ ( x ) = − 0.08 x + 24 = 0 , which gives x = 300 .
Verify that this is a maximum by checking the second derivative P ′′ ( x ) = − 0.08 < 0 .
The production level that maximizes profit is 300 .
Explanation
Problem Setup We are given the cost function C ( x ) = 1000 + 8 x + 0.04 x 2 and the price function p ( x ) = 32 . We want to find the production level x that maximizes profit.
Revenue Function The revenue function R ( x ) is the product of the price and the quantity, so R ( x ) = x "." p ( x ) = 32 x .
Profit Function The profit function P ( x ) is the difference between the revenue and the cost, so P ( x ) = R ( x ) − C ( x ) = 32 x − ( 1000 + 8 x + 0.04 x 2 ) = − 0.04 x 2 + 24 x − 1000 .
Finding Critical Points To maximize the profit, we need to find the critical points of the profit function. We take the derivative of the profit function with respect to x and set it equal to zero: P ′ ( x ) = − 0.08 x + 24 = 0 .
Solving for x Solving for x , we get 0.08 x = 24 , so x = 0.08 24 = 300 .
Verifying Maximum To confirm that this is a maximum, we can check the second derivative of the profit function: P ′′ ( x ) = − 0.08 . Since the second derivative is negative, the profit function is concave down, and x = 300 is indeed a maximum.
Final Answer Therefore, the production level that will maximize the profit is x = 300 .
Examples
Consider a small business producing handmade goods. The cost function includes fixed costs (rent, utilities) and variable costs (materials, labor). The price function determines the revenue per item sold. By finding the production level that maximizes profit, the business can optimize its operations to achieve the highest possible earnings. This ensures efficient resource allocation and sustainable growth.
