Homework SourseA manufacturer is planning to sell a new product at the pric
A manufacturer is planning to sell a new product at the price of $260 per unit and estimates that if x thousand dollars is spent on development and y thousand dollars is spent on promotion, consumers will buy approximately ((200y)/(y+1))+((250x)/(x+4)) units of the product. If manufacturing costs $120 per unit, how much should the manufacturer spend on development and how much on promotion to generate the largest possible profit from the sale of this product?
Promotion:____________ dollars?
Development:_______________ dollars?
Promotion:____________ dollars?
Development:_______________ dollars?
Solution
Profit P = 260*((200y)/(y+1))+((250x)/(x+4)) - 1000*x - 1000*y - 120*((200y)/(y+1))+((250x)/(x+4))
P = 140*((200y)/(y+1))+((250x)/(x+4)) - 1000*(x+y)
dP/dx = 140*250*4/(x+4)2 - 1000
dP/dy = 140*200/(y+1)2 - 1000
For max. profit, dP/dx = 0 and dP/dy = 0.
Therefore, 140*250*4/(x+4)2 - 1000 = 0 and 140*200/(y+1)2 - 1000 = 0.
This gives, x = 7.832 and y = 4.292
Development = 1000*7.832 = $7832
Promotion = 1000*4.292 = $4292
