A manager has been allotted $7000 to spend on the development and promotion of a

Question

A manager has been allotted $7000 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) = 48x^1/2 y^3/2 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? Development dollars Promotion dollars Suppose the editor is allotted an extra s1000 for development and promotion. Use the Lagrange multiplier lambda to estimate the change in the maximum sales level. Increase of units sold

Explanation / Answer

Solution:

a)
f (x,y) = 48x1/2y3/2 = 48(xy3)
x + y = 7000
f (x, y, ) = 48(xy3) - (x + y - 7000) = 0

fx = (48y3)/(2x) - = 0
= (48y3)/(2x)

fy = 72(xy) - = 0
= 72(xy)

=> 72(xy) = (48y3)/(2x)
(3/2) (xy) = (y3)/(2x)
[(3/2) (xy)]2 = [(y3)/(2x)]2
(9/4) xy = y3/(4x)
9x2 = y2
y = 3x
``````````````
f = x + y - 7000 = 0
x + y - 7000 = 0
x + (3x) - 7000 = 0
4x = 7000
x = $1750 (development)

y = 3x
y = 3(1750)
y = $5250 (Promotion)

b)
= 72(xy)
= 72(1750*5250)
= $218238
increases by $218238

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