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)

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)​

Quizplus · Accepted Answer

To find the maximum profit, we need to find the production level that maximizes the profit. Let P(x) be the profit function. Then, 
P(x) = R(x) - C(x) 
where R(x) is the revenue function and C(x) is the cost function. 
Since the market price is $62 per unit, the revenue function is given by 
R(x) = 62x 
The cost function is given by 
C(x) = 11eb0d3b_986a_a6ec_9790_f77d71c97185_TB1243_11
= 11eb0d3b_986a_a6ed_9790_c119b0dcf628_TB1243_11 
Therefore, the profit function is 
P(x) = 62x - 11eb0d3b_986a_a6ed_9790_c119b0dcf628_TB1243_11 
To find the production level that maximizes the profit, we need to find the critical points of the profit function. 
P'(x) = 62 - 11eb0d3b_986a_a6ed_9790_c119b0dcf628_TB1243_11 
Setting P'(x) = 0 and solving for x, we get 
x = (11eb0d3b_986a_a6ed_9790_c119b0dcf628_TB1243_11)/62 = 392.9032258 
Since the production is limited to 430 units per week, the maximum production level is 392 units per week. 
Therefore, the maximum profit is 
P(392) = 62(392) - 11eb0d3b_986a_a6ed_9790_c119b0dcf628_TB1243_11 
= 11eb0d3b_986a_ce00_9790_61cd31160663_TB1243_11.

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