Question

A firm has monthly average costs, in dollars, given by C=42000/x + 200 + x

where x is the number of units produced per month. The firm can sell its product in a competitive market for $2100 per unit. If production is limited to 700 units per month, find the number of units that gives maximum profit. X= 700 units

a) FIND THE MAXIMUM PROFIT

Answer 1

To find the number of units that gives maximum profit, we need to determine the production level that maximizes the difference between revenue and cost. The maximum profit is achieved at 10 units.

Step-by-step explanation:

To find the number of units that gives maximum profit, we need to determine the production level that maximizes the difference between revenue and cost. In this case, revenue is given by the selling price of the product multiplied by the number of units produced. Cost is given by the fixed costs plus the variable costs per unit. So we have:

R = 2100x, where R is revenue

C = 42000/x + 200 + x, where C is cost

To find the maximum profit, we need to find the value of x that maximizes the difference between R and C. We can do this by taking the derivative of the profit function, setting it equal to zero, and solving for x.

Let P = R - C, where P is profit

Differentiate P with respect to x:

P' = 2100 - (42000/x^2 + 1)

Set P' equal to zero and solve for x:

2100 - (42000/x^2 + 1) = 0

42000/x^2 + 1 = 2100

42000/x^2 = 2099

x^2 = 42000/2099

x ~ 10.148

Since the number of units produced must be a whole number, the maximum profit is achieved at x = 10.

Therefore, the number of units that gives maximum profit is 10 units.