A manager has been allotted $10000 to spend on the development and promotion of

Question

A manager has been allotted $10000 to spend on the development and promotion of a new product. It is estimated that if x thousand dollars are spent on development and y thousand dollars on promotion, approximately f(x,y)=85x^(1/2)y^(3/2) units of the product will be sold.

a. How much money should the editor allocate to development and how much to promotion in order to maximize sales?

Development: dollars
Promotion:      dollars

b. Suppose the editor is allotted an extra $1000 for development and promotion. Use the Lagrange multiplier ? to estimate the change in the maximum sales level.

Increase of:    units sold

Explanation / Answer

Part b:
f(x,y)=86x^(1/2)y^(3/2)
The constraint g(x,y) = x+y = 8000
The gradient of f = <43x^(-1/2)y^(3/2), 129x^(1/2)y^(1/2)>
The gradient of g = <1,1>
Setting f = Lg:
43x^(-1/2)y^(3/2) = 129x^(1/2)y^(1/2)
y^(3/2) = 3xy^(1/2)
y^(1/2)(y-3x) =0
y = 0 or y=3x
1. y = 0 -> x = 8000
2. y = 3x
x + 3x = 8000
x = 2000, y = 6000
f(x,y) = 86x^(1/2)y^(3/2)
= 86 * (2000)^(1/2) * 6000^(3/2)
= 86 * 20sqrt(5) * 120000sqrt(15)
= 1032000000sqrt(3) ------ max units sold
Now calculate the increase of units sold

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