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 $210 per unit and estimates that if thousand dollars is spent on development and y thousand dollars is spent on promotion, consumers will buy approximately (300y/(y+5))+(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 = 210*(300y/(y+5))+(250x/(x+4)) - 1000*x - 1000*y - 120*(300y/(y+5))+(250x/(x+4))
P = 90*(300y/(y+5))+(250x/(x+4)) - 1000*(x+y)
dP/dx = 90*250*4/(x+4)2 - 1000
dP/dy = 90*300*5/(y+5)2 - 1000
For maximum profit, dP/dx = 0 and dP/dy = 0.
Hence, 90*250*4/(x+4)2 - 1000 = 0 and 90*300*5/(y+5)2 - 1000 = 0
Hence, x = 5.487 and y = 6.619
So, promotion = 1000*6.619 = $6619
Development = 1000*5.487 = $5487
