Hi,, Q1) A manufacturer is planning to sell a new product at the price of $230 p

Question

Hi,,

Q1) A manufacturer is planning to sell a new product at the price of $230 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 [(350y)/(y+2)] + [(50x)/(x+3)] units of the product. If manufacturing costs $130 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: $

Development: $

Absolute minimum value: attained at

Absolute maximum value: attained at

In each case, your second answer should be one or more points (X,Y). If there is more than one point at which the maximum or minimum value is attained, enter them all separated by commas (e.g. (1,2),(1,3) ).

Explanation / Answer

I get optimum values for x and y of

x = 10?(15) - 3 ? 35.73 and y = 10?(35) - 1 ? 58.16

If z units are purchased, then revenue is 230z. The manufacturing cost would be 130z. The total cost would be

total cost = manufacturing cost + promotion cost + development cost

= 130z + y + x.

This makes profit

P = revenue - total cost = 230z - 130z - y - x = 100z - x - y.

We're given that z = 350y/(y + 1) + 50x/(x + 3). The function to maximize is therefore

P(x, y) = 3500y/(y + 1) + 500x/(x + 3) - y - x.

Its gradient is

?P(x, y) = <1500/(x + 3)

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