A small business has weekly average costs (in dollars) of , where x is the number of units produced each week. The competitive market price for this business's product is $62 per unit. If production is limited to 430 units per week, find the maximum profit. (Note: )
C)
D)
E)

Question

A small business has weekly average costs (in dollars) of  , where x is the number of units produced each week. The competitive market price for this business's product is $62 per unit. If production is limited to 430 units per week, find the maximum profit. (​Note:  ) ​​
C) ​ 
D) ​ 
E) ​

Quizplus · Accepted Answer

The profit function can be expressed as P(x) = R(x) - C(x), where R(x) is the revenue function and C(x) is the cost function.

In this case, the revenue function is R(x) = 62x, since the product sells for $62 per unit and x is the number of units produced.

The cost function is C(x) = 11ea984b_b241_3b4f_9029_1fbe260b6e8b_TB4005_11, as given in the problem.

Therefore, the profit function can be expressed as:

P(x) = 62x - 11ea984b_b241_3b4f_9029_1fbe260b6e8b_TB4005_11

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

P'(x) = 62 - 11ea984b_b241_3b4f_9029_1fbe260b6e8b_TB4005_11 = 0

Solving for x gives:

x = 430

We can verify that this is a maximum by taking the second derivative of the profit function and evaluating it at x = 430:

P''(x) = 0

Since the second derivative is zero, we cannot determine whether the function is concave up or down at x = 430. However, since P'(x) changes from positive to negative at x = 430, we know that this is a maximum.

Substituting x = 430 into the profit function gives:

P(430) = 62(430) - 11ea984b_b241_3b4f_9029_1fbe260b6e8b_TB4005_11 = 11ea984b_b241_8973_9029_bb1bc02c0791_TB4005_11

Therefore, the maximum profit is 11ea984b_b241_8973_9029_bb1bc02c0791_TB4005_11, which corresponds to choice C.

A Small Business Has Weekly Average Costs (In Dollars) of