A manager has been allotted $5000 to spend on the development and promotion of a

Question

A manager has been allotted $5000 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)=79x^(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?

Answer to part A is development=1250 promotion=3750

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 how many units sold?

Explanation / Answer

Here   we will solve it by   using   lagrange method

The constraint g(x,y) = x+y = 5000

he gradient of f = 39.5x^(-1/2)y^(3/2), 118.5x^(1/2)y^(1/2)

The gradient of g = <1,1>
Setting f = Lg:
39.5x^(-1/2)y^(3/2) = 118,5x^(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 = 5000
2. y = 3x
x + 3x = 5000
x = 1250, y = 3750

b)f(x,y) = 79x^(1/2)y^(3/2)
= 79 * (1250)^(1/2) * 3750^(3/2)
= 79 * 25sqrt(2) * 3750*25sqrt(6)
= 379,312,500sqrt(3) ------ max units sold
Now calculate the increase of units sold

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.