A manager has been allotted $11000 to spend on the development and promotion of

Question

A manager has been allotted $11000 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)=66x1/2y3/2 units of the product will be sold

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

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.

Explanation / Answer

a) We want to maximize f(x,y) = 66x^(1/2)y^(3/2), subject to the constraint g(x,y) = x + y = 11 (since x, y are measured in thousands).

By Lagrange Multipliers, ?f = ??g.
==> <33x^(-1/2)y^(3/2), 99x^(1/2)y^(1/2)> = ?<1, 1>.

Equate like entries:
33x^(-1/2)y^(3/2) = ? = 99x^(1/2)y^(1/2)
==> y = 3x.

Substitute this into g:
x + 3x = 11 ==> x = 11/4.

So, (x, y) = (11/4, 33/4).
Development: 11/4 * 1000 = 2750 dollars
Production: 33/4 * 1000 = 8250 dollars.
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b) ? approximates the change of the maximum value resulting in a one unit increase of g (from 11 to 11+1 = 12).

From our work in part a,
? = 99 (11/4)^(1/2) (33/4)^(1/2)
...= 471.55 units = 472 units (round to nearest integer).

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