Multivariable optimiziation?

I have an equation that is rather unwieldy, but can be simplified greatly to k=2n*x*y/(x+y) through substitution of its parts. I have calculated the partial derivatives of k with respect to n, x, and y, and they are the same as the partial derivatives of the original equation (with the exception of a constant factor of 1/100 for the partial derivative of n), and plugging the substitutions back into the equations has given me a set of related rates. I can tell how quickly k increases by increasing any of the other three variables at any one point.

The question I h

Question

Multivariable optimiziation?

I have an equation that is rather unwieldy, but can be simplified greatly to k=2n*x*y/(x+y) through substitution of its parts. I have calculated the partial derivatives of k with respect to n, x, and y, and they are the same as the partial derivatives of the original equation (with the exception of a constant factor of 1/100 for the partial derivative of n), and plugging the substitutions back into the equations has given me a set of related rates. I can tell how quickly k increases by increasing any of the other three variables at any one point.

The question I h

anonymous · Answer

Ah, size limits. I'll post the rest of it.

anonymous · Answer

The question I have is, if I wanted to substitute the costs of each unit (to be precise, each unit of n has a cost of $2.6, each unit of x has a cost of $15.6, and each unit of y has a cost of $15.4) to calculate the most cost-efficient way of increasing k at any given moment, how would I do that?

anonymous · Answer

Original Equation: k=(N/50)(X+100)(Y+100)/(X+Y+200)
Substitute
n=N/100
x=X+100
y=Y+100
New equation: k=2nxy/(x+y)
which can also be written as the harmonic mean m of x and y
Mean equation: k=n*m

anonymous · Answer

dk/dn = (2x*y)/(x+y), but this is off by a factor of 1/100, so
dk/dn = (x*y)/(50(x+y))
dk/dx = (2n*y^2)/(x+y)^2
dk/dy = (2n*x^2)/(x+y)^2
Substitute for original values of x, y, and n
dk/dn = ((X+100)(Y+100))/(50(X+Y+200))
dk/dx = (2*(N/100)*(Y+100)^2)/(X+Y+200)^2
dk/dy = (2*(N/100)*(X+100)^2)/(X+Y+200)^2

anonymous · Answer

Cost of one unit of N = $2.60
Cost of one unit of X = $15.60
Cost of one unit of Y = $15.40

I understand that each partial derivative is a rate of change for k, but if you just look at the output it almost always appears to be most efficient to purchase X or Y, but this is not the case due to their higher costs. How do I use the costs of X, Y, and N to find out which is most efficient to purchase at any given moment to increase k?

anonymous · Answer

I have also considered making the cost of X and Y identical at $15.50 for simplicity. I am unsure how greatly it will impact the results if I leave them as they are.