Question

If the fixed costs are 25 , variable costs per unit are 2 and the price function is p(x)=20−x,x is the quantity of product. 1. What is a profit function? (2marks) 2. Find the levels of output which give a profit of 31 . (2 marks) 3. Find the maximum profit and the value of x at which it is achieved. (2 marks)

Answer 1

The profit function is calculated as total revenue minus total costs. To find a specific profit level, set the profit equation equal to that value and solve for x. The maximum profit in the given scenario is $40 at an output of 5 units.

Step-by-step explanation:

The profit function is the total revenue minus the total costs. For a given price function p(x) = 20 - x, where x is the quantity of product, the profit function is the revenue (p(x) * x) minus the fixed costs and the variable costs (2x). Fixed costs are given as 25, so the profit function would be π(x) = x(20 - x) - 25 - 2x = 20x - x² - 2x - 25.

To find the levels of output which give a profit of 31, we need to solve the equation 20x - x² - 2x - 25 = 31, which simplifies to x² - 18x + 56 = 0. Solving this quadratic equation yields the values of x that give the specified profit.

For the maximum profit and the value of x at which it is achieved, we can use calculus to find the derivative of the profit function and set it to zero, or simply examine the profit functions values at different quantities. In this case, it is given that the profit-maximizing output level is 5, with a profit of $40.