A company produces and sells a comsumer product and is able to contral the demand for the product by varying the selling price.The approximate relationship between price and demand is
p=$40+(3800/D)-(6000/D^2) , for D>1 ,
where p is the price per unit in dollars and D is the demand per month .The company is seeking to maximize its profit.The fixed cost is $1500 per month and the variable cost(cv) is $42 per unit.
(1)What is the number of units that should be produced and sold each month to maximize profit?
(2) Show that your answer to part(1) maximize profit.
- 小白Lv 78 years agoFavorite Answer
價格 p = 40 + 3800/D - 6000/D^2
數量 = D
因此，營業額 = D × p = 40D + 3800 - 6000/D
總成本 = 1500 + 42D
利潤 = 營業額 - 總成本 = 40D + 3800 - 6000/D - 1500 - 42D = 2300 - (2D + 6000/D)
算術平均大於等於幾何平均 ==> 2D + 6000/D ≥ 2 × √(2D × 6000/D) = 219
因此，利潤 ≤ 2081
此時，2D = 6000/D ==> D = 54 或 55
D = 54，利潤 = 2081
D = 55，利潤 = 2081
生產 54 或 55 個，可達最大利潤。