A company has $27600 to spend on the development and promotion of a new product.

Question

The company estimates that if x is spent on development and y is spent on promotion, then approximately (x^3/2*y^1/2)/80000 items of new product will be sold. Based on this estimate, how much should the company spend on development so the maximum number of products can be sold?

Answer 1

This is another, slightly different way to solve it:

product sold (p) = (x^3/2*y^1/2)/80000 -----(1)

the money available for development and production = x + y = 27600 ----(2)

diffentiating (1) on both sides

dp/dx = y^1/2 (3/2*x^1/2) + x^3/2 * (1/2 * y^-1/2) dy/dx ----(3)

diffentiating (2) on both sides

1+dy/dx=0

dy/dx=-1 ---- (4)

substituting (4) in (3),

dp/dx=3/2 (xy)^1/2 - 1/2 x^3/2 y^-1/2

when p is maximum, dp/dx = 0

3/2 (xy)^1/2 - 1/2 x^3/2 y^-1/2 = 0

1/2 (xy)^1/2 (3 - x/y)= 0

3-x/y = 0

y = x/3

x + y = 27600

x + x/3 = 27600; x = 27600*3/4 = 20700

Answer 2

P(x) = (x^3/2 - y^1/2) / 8000

8000 P(x) = (x^3/2 * y^1/2), y = 27600 - x

8000 P(x) = (x^3/2)(27600 - x)^1/2

64000 P(x) = x³(27600 - x)

Can you isolate P(x), take the first derivative, and answer?