(1 pt) A manager has been allotted $8000 to spend on the development and promoti

Question

(1 pt) A manager has been allotted $8000 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)=100x^(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

I already get the answer of a, which is 2000 and 6000

But I can't get the correct answer of b, my answer is 100(20+2.5)^(1/2)(60+7.5)^(3/2)-100(20)^(1/2)(60)^(3/2) which is 55209.119491258. but it's wrong, and I don't know where did I made the mistake.

Explanation / Answer

I am solving only part b

f(x,y)=100x^(1/2)y^(3/2)
The constraint g(x,y) = x+y = 9000
The gradient of f = <50x^(-1/2)y^(3/2), 150x^(1/2)y^(1/2)>
The gradient of g = <1,1>
Setting f = Lg:
100x^(-1/2)y^(3/2) = 150x^(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 = 9000
2. y = 3x
x + 3x = 9000
x = 2250, y = 6750
f(x,y) = 100x^(1/2)y^(3/2)
= 100 * (2250)^(1/2) * 6750^(3/2)
= 100*15sqrt(10)*15*sqrt(30)
= 225000sqrt(3) ------ max units sold
= 389711.4317 ------ max units sold
Now calculate the increase of units sold

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